diff --git a/spec/unit/models/MSC3089Branch.spec.ts b/spec/unit/models/MSC3089Branch.spec.ts
index d782a4276..1daf1c07f 100644
--- a/spec/unit/models/MSC3089Branch.spec.ts
+++ b/spec/unit/models/MSC3089Branch.spec.ts
@@ -312,7 +312,7 @@ describe("MSC3089Branch", () => {
         } as MatrixEvent);
 
         const events = [await branch.getFileEvent(), await branch2.getFileEvent(), {
-            replacingEventId: () => null,
+            replacingEventId: (): string => null,
             getId: () => "$unknown",
         }];
         staticRoom.getLiveTimeline = () => ({ getEvents: () => events }) as EventTimeline;
diff --git a/src/@types/global.d.ts b/src/@types/global.d.ts
index 0a857b4e8..c7e65ff77 100644
--- a/src/@types/global.d.ts
+++ b/src/@types/global.d.ts
@@ -62,12 +62,6 @@ declare global {
         };
     }
 
-    interface HTMLAudioElement {
-        // sinkId & setSinkId are experimental and typescript doesn't know about them
-        sinkId: string;
-        setSinkId(outputId: string);
-    }
-
     interface DummyInterfaceWeShouldntBeUsingThis {}
 
     interface Navigator {
diff --git a/src/client.ts b/src/client.ts
index f9a12f332..1171ddbee 100644
--- a/src/client.ts
+++ b/src/client.ts
@@ -621,7 +621,7 @@ export interface IMyDevice {
     last_seen_ts?: number;
 }
 
-interface IDownloadKeyResult {
+export interface IDownloadKeyResult {
     failures: { [serverName: string]: object };
     device_keys: {
         [userId: string]: {
@@ -632,13 +632,42 @@ interface IDownloadKeyResult {
             };
         };
     };
+    // the following three fields were added in 1.1
+    master_keys?: {
+        [userId: string]: {
+            keys: { [keyId: string]: string };
+            usage: string[];
+            user_id: string;
+        };
+    };
+    self_signing_keys?: {
+        [userId: string]: {
+            keys: { [keyId: string]: string };
+            signatures: ISignatures;
+            usage: string[];
+            user_id: string;
+        };
+    };
+    user_signing_keys?: {
+        [userId: string]: {
+            keys: { [keyId: string]: string };
+            signatures: ISignatures;
+            usage: string[];
+            user_id: string;
+        };
+    };
 }
 
-interface IClaimOTKsResult {
+export interface IClaimOTKsResult {
     failures: { [serverName: string]: object };
     one_time_keys: {
         [userId: string]: {
-            [deviceId: string]: string;
+            [deviceId: string]: {
+                [keyId: string]: {
+                    key: string;
+                    signatures: ISignatures;
+                };
+            };
         };
     };
 }
@@ -1421,14 +1450,12 @@ export class MatrixClient extends EventEmitter {
             }
         }
 
-        // We swallow errors because we need a default object anyhow
         return this.http.authedRequest(
             undefined, "GET", "/capabilities",
-        ).catch((e: Error) => {
+        ).catch((e: Error): void => {
+            // We swallow errors because we need a default object anyhow
             logger.error(e);
-            return null; // otherwise consume the error
-        }).then((r) => {
-            if (!r) r = {};
+        }).then((r: { capabilities?: ICapabilities } = {}) => {
             const capabilities: ICapabilities = r["capabilities"] || {};
 
             // If the capabilities missed the cache, cache it for a shorter amount
@@ -2836,7 +2863,7 @@ export class MatrixClient extends EventEmitter {
         }
 
         let totalKeyCount = 0;
-        let keys = [];
+        let keys: IMegolmSessionData[] = [];
 
         const path = this.makeKeyBackupPath(targetRoomId, targetSessionId, backupInfo.version);
 
@@ -5692,18 +5719,14 @@ export class MatrixClient extends EventEmitter {
         searchResults.count = roomEvents.count;
         searchResults.next_batch = roomEvents.next_batch;
 
-        // combine the highlight list with our existing list; build an object
-        // to avoid O(N^2) fail
-        const highlights = {};
-        roomEvents.highlights.forEach((hl) => {
-            highlights[hl] = 1;
-        });
+        // combine the highlight list with our existing list;
+        const highlights = new Set<string>(roomEvents.highlights);
         searchResults.highlights.forEach((hl) => {
-            highlights[hl] = 1;
+            highlights.add(hl);
         });
 
         // turn it back into a list.
-        searchResults.highlights = Object.keys(highlights);
+        searchResults.highlights = Array.from(highlights);
 
         // append the new results to our existing results
         const resultsLength = roomEvents.results ? roomEvents.results.length : 0;
@@ -5794,11 +5817,9 @@ export class MatrixClient extends EventEmitter {
 
         return this.http.authedRequest(
             undefined, "GET", path, undefined, undefined,
-        ).then((response) => {
+        ).then((response: IFilterDefinition) => {
             // persist the filter
-            const filter = Filter.fromJson(
-                userId, filterId, response,
-            );
+            const filter = Filter.fromJson(userId, filterId, response);
             this.store.storeFilter(filter);
             return filter;
         });
@@ -6096,7 +6117,7 @@ export class MatrixClient extends EventEmitter {
             {
                 prefix: '',
             },
-        ).catch((e) => {
+        ).catch((e: Error) => {
             // Need to unset this if it fails, otherwise we'll never retry
             this.serverVersionsPromise = null;
             // but rethrow the exception to anything that was waiting
@@ -6420,7 +6441,7 @@ export class MatrixClient extends EventEmitter {
     public isUsernameAvailable(username: string): Promise<true> {
         return this.http.authedRequest(
             undefined, "GET", '/register/available', { username: username },
-        ).then((response) => {
+        ).then((response: { available: boolean }) => {
             return response.available;
         });
     }
@@ -7741,11 +7762,11 @@ export class MatrixClient extends EventEmitter {
      *     an error response ({@link module:http-api.MatrixError}).
      */
     public claimOneTimeKeys(
-        devices: string[],
+        devices: [string, string][],
         keyAlgorithm = "signed_curve25519",
         timeout?: number,
     ): Promise<IClaimOTKsResult> {
-        const queries = {};
+        const queries: Record<string, Record<string, string>> = {};
 
         if (keyAlgorithm === undefined) {
             keyAlgorithm = "signed_curve25519";
diff --git a/src/crypto/CrossSigning.ts b/src/crypto/CrossSigning.ts
index 8eda069a4..077d705b8 100644
--- a/src/crypto/CrossSigning.ts
+++ b/src/crypto/CrossSigning.ts
@@ -33,6 +33,7 @@ import { OlmDevice } from "./OlmDevice";
 import { ICryptoCallbacks } from "../matrix";
 import { ISignatures } from "../@types/signed";
 import { CryptoStore } from "./store/base";
+import { ISecretStorageKeyInfo } from "./api";
 
 const KEY_REQUEST_TIMEOUT_MS = 1000 * 60;
 
@@ -175,7 +176,7 @@ export class CrossSigningInfo extends EventEmitter {
         // check what SSSS keys have encrypted the master key (if any)
         const stored = await secretStorage.isStored("m.cross_signing.master", false) || {};
         // then check which of those SSSS keys have also encrypted the SSK and USK
-        function intersect(s) {
+        function intersect(s: Record<string, ISecretStorageKeyInfo>) {
             for (const k of Object.keys(stored)) {
                 if (!s[k]) {
                     delete stored[k];
diff --git a/src/crypto/DeviceList.ts b/src/crypto/DeviceList.ts
index c8a87cc56..6e951263c 100644
--- a/src/crypto/DeviceList.ts
+++ b/src/crypto/DeviceList.ts
@@ -28,7 +28,7 @@ import { CrossSigningInfo, ICrossSigningInfo } from './CrossSigning';
 import * as olmlib from './olmlib';
 import { IndexedDBCryptoStore } from './store/indexeddb-crypto-store';
 import { chunkPromises, defer, IDeferred, sleep } from '../utils';
-import { MatrixClient } from "../client";
+import { IDownloadKeyResult, MatrixClient } from "../client";
 import { OlmDevice } from "./OlmDevice";
 import { CryptoStore } from "./store/base";
 
@@ -756,17 +756,21 @@ class DeviceListUpdateSerialiser {
             opts.token = this.syncToken;
         }
 
-        const factories = [];
+        const factories: Array<() => Promise<IDownloadKeyResult>> = [];
         for (let i = 0; i < downloadUsers.length; i += this.deviceList.keyDownloadChunkSize) {
             const userSlice = downloadUsers.slice(i, i + this.deviceList.keyDownloadChunkSize);
             factories.push(() => this.baseApis.downloadKeysForUsers(userSlice, opts));
         }
 
-        chunkPromises(factories, 3).then(async (responses: any[]) => {
-            const dk = Object.assign({}, ...(responses.map(res => res.device_keys || {})));
-            const masterKeys = Object.assign({}, ...(responses.map(res => res.master_keys || {})));
-            const ssks = Object.assign({}, ...(responses.map(res => res.self_signing_keys || {})));
-            const usks = Object.assign({}, ...(responses.map(res => res.user_signing_keys || {})));
+        chunkPromises(factories, 3).then(async (responses: IDownloadKeyResult[]) => {
+            const dk: IDownloadKeyResult["device_keys"]
+                = Object.assign({}, ...(responses.map(res => res.device_keys || {})));
+            const masterKeys: IDownloadKeyResult["master_keys"]
+                = Object.assign({}, ...(responses.map(res => res.master_keys || {})));
+            const ssks: IDownloadKeyResult["self_signing_keys"]
+                = Object.assign({}, ...(responses.map(res => res.self_signing_keys || {})));
+            const usks: IDownloadKeyResult["user_signing_keys"]
+                = Object.assign({}, ...(responses.map(res => res.user_signing_keys || {})));
 
             // yield to other things that want to execute in between users, to
             // avoid wedging the CPU
@@ -811,8 +815,12 @@ class DeviceListUpdateSerialiser {
 
     private async processQueryResponseForUser(
         userId: string,
-        dkResponse: object,
-        crossSigningResponse: any, // TODO types
+        dkResponse: IDownloadKeyResult["device_keys"]["user_id"],
+        crossSigningResponse: {
+            master: IDownloadKeyResult["master_keys"]["user_id"];
+            self_signing: IDownloadKeyResult["master_keys"]["user_id"]; // eslint-disable-line camelcase
+            user_signing: IDownloadKeyResult["user_signing_keys"]["user_id"]; // eslint-disable-line camelcase
+        },
     ): Promise<void> {
         logger.log('got device keys for ' + userId + ':', dkResponse);
         logger.log('got cross-signing keys for ' + userId + ':', crossSigningResponse);
@@ -869,7 +877,7 @@ async function updateStoredDeviceKeysForUser(
     olmDevice: OlmDevice,
     userId: string,
     userStore: Record<string, DeviceInfo>,
-    userResult: object,
+    userResult: IDownloadKeyResult["device_keys"]["user_id"],
     localUserId: string,
     localDeviceId: string,
 ): Promise<boolean> {
diff --git a/src/crypto/backup.ts b/src/crypto/backup.ts
index 161e7b40f..295b48e9f 100644
--- a/src/crypto/backup.ts
+++ b/src/crypto/backup.ts
@@ -33,6 +33,7 @@ import { encryptAES, decryptAES, calculateKeyCheck } from './aes';
 import { getCrypto } from '../utils';
 import { ICurve25519AuthData, IAes256AuthData, IKeyBackupInfo, IKeyBackupSession } from "./keybackup";
 import { UnstableValue } from "../NamespacedValue";
+import { IMegolmSessionData } from "./index";
 
 const KEY_BACKUP_KEYS_PER_REQUEST = 200;
 
@@ -87,7 +88,7 @@ interface BackupAlgorithmClass {
 interface BackupAlgorithm {
     untrusted: boolean;
     encryptSession(data: Record<string, any>): Promise<any>;
-    decryptSessions(ciphertexts: Record<string, IKeyBackupSession>): Promise<Record<string, any>[]>;
+    decryptSessions(ciphertexts: Record<string, IKeyBackupSession>): Promise<IMegolmSessionData[]>;
     authData: AuthData;
     keyMatches(key: ArrayLike<number>): Promise<boolean>;
     free(): void;
@@ -300,7 +301,7 @@ export class BackupManager {
         const ret = {
             usable: false,
             trusted_locally: false,
-            sigs: [],
+            sigs: [] as SigInfo[],
         };
 
         if (
@@ -320,7 +321,7 @@ export class BackupManager {
             ret.trusted_locally = true;
         }
 
-        const mySigs = backupInfo.auth_data.signatures[this.baseApis.getUserId()] || [];
+        const mySigs = backupInfo.auth_data.signatures[this.baseApis.getUserId()] || {};
 
         for (const keyId of Object.keys(mySigs)) {
             const keyIdParts = keyId.split(':');
@@ -645,9 +646,7 @@ export class Curve25519 implements BackupAlgorithm {
         return this.publicKey.encrypt(JSON.stringify(plainText));
     }
 
-    public async decryptSessions(
-        sessions: Record<string, IKeyBackupSession>,
-    ): Promise<Record<string, any>[]> {
+    public async decryptSessions(sessions: Record<string, IKeyBackupSession>): Promise<IMegolmSessionData[]> {
         const privKey = await this.getKey();
         const decryption = new global.Olm.PkDecryption();
         try {
@@ -658,7 +657,7 @@ export class Curve25519 implements BackupAlgorithm {
                 throw { errcode: MatrixClient.RESTORE_BACKUP_ERROR_BAD_KEY };
             }
 
-            const keys = [];
+            const keys: IMegolmSessionData[] = [];
 
             for (const [sessionId, sessionData] of Object.entries(sessions)) {
                 try {
@@ -777,8 +776,8 @@ export class Aes256 implements BackupAlgorithm {
         return await encryptAES(JSON.stringify(plainText), this.key, data.session_id);
     }
 
-    async decryptSessions(sessions: Record<string, IKeyBackupSession>): Promise<Record<string, any>[]> {
-        const keys = [];
+    async decryptSessions(sessions: Record<string, IKeyBackupSession>): Promise<IMegolmSessionData[]> {
+        const keys: IMegolmSessionData[] = [];
 
         for (const [sessionId, sessionData] of Object.entries(sessions)) {
             try {
diff --git a/src/crypto/dehydration.ts b/src/crypto/dehydration.ts
index c9a9211fa..652f72e3e 100644
--- a/src/crypto/dehydration.ts
+++ b/src/crypto/dehydration.ts
@@ -22,9 +22,7 @@ import { decryptAES, encryptAES } from './aes';
 import { logger } from '../logger';
 import { ISecretStorageKeyInfo } from "./api";
 import { Crypto } from "./index";
-
-// FIXME: these types should eventually go in a different file
-type Signatures = Record<string, Record<string, string>>;
+import { ISignatures } from "../@types/signed";
 
 export interface IDehydratedDevice {
     device_id: string; // eslint-disable-line camelcase
@@ -43,13 +41,13 @@ export interface IDeviceKeys {
     device_id: string; // eslint-disable-line camelcase
     user_id: string; // eslint-disable-line camelcase
     keys: Record<string, string>;
-    signatures?: Signatures;
+    signatures?: ISignatures;
 }
 
 export interface IOneTimeKey {
     key: string;
     fallback?: boolean;
-    signatures?: Signatures;
+    signatures?: ISignatures;
 }
 
 export const DEHYDRATION_ALGORITHM = "org.matrix.msc2697.v1.olm.libolm_pickle";
@@ -244,7 +242,7 @@ export class DehydrationManager {
             }
 
             logger.log("Preparing one-time keys");
-            const oneTimeKeys = {};
+            const oneTimeKeys: Record<string, IOneTimeKey> = {};
             for (const [keyId, key] of Object.entries(otks.curve25519)) {
                 const k: IOneTimeKey = { key };
                 const signature = account.sign(anotherjson.stringify(k));
diff --git a/src/crypto/index.ts b/src/crypto/index.ts
index 6b71ed8ec..85ba4de45 100644
--- a/src/crypto/index.ts
+++ b/src/crypto/index.ts
@@ -179,6 +179,13 @@ export interface IEventDecryptionResult {
     untrusted?: boolean;
 }
 
+export interface IRequestsMap {
+    getRequest(event: MatrixEvent): VerificationRequest;
+    getRequestByChannel(channel: IVerificationChannel): VerificationRequest;
+    setRequest(event: MatrixEvent, request: VerificationRequest): void;
+    setRequestByChannel(channel: IVerificationChannel, request: VerificationRequest): void;
+}
+
 export class Crypto extends EventEmitter {
     /**
      * @return {string} The version of Olm.
@@ -808,7 +815,7 @@ export class Crypto extends EventEmitter {
             }
         };
 
-        const signKeyBackupWithCrossSigning = async (keyBackupAuthData) => {
+        const signKeyBackupWithCrossSigning = async (keyBackupAuthData: IKeyBackupInfo["auth_data"]) => {
             if (
                 this.crossSigningInfo.getId() &&
                 await this.crossSigningInfo.isStoredInKeyCache("master")
@@ -1148,7 +1155,7 @@ export class Crypto extends EventEmitter {
         const signedDevice = await this.crossSigningInfo.signDevice(this.userId, device);
         logger.info(`Starting background key sig upload for ${this.deviceId}`);
 
-        const upload = ({ shouldEmit }) => {
+        const upload = ({ shouldEmit = false }) => {
             return this.baseApis.uploadKeySignatures({
                 [this.userId]: {
                     [this.deviceId]: signedDevice,
@@ -1184,7 +1191,7 @@ export class Crypto extends EventEmitter {
 
             // Check all users for signatures if upgrade callback present
             // FIXME: do this in batches
-            const users = {};
+            const users: Record<string, IDeviceVerificationUpgrade> = {};
             for (const [userId, crossSigningInfo]
                 of Object.entries(this.deviceList.crossSigningInfo)) {
                 const upgradeInfo = await this.checkForDeviceVerificationUpgrade(
@@ -1482,7 +1489,7 @@ export class Crypto extends EventEmitter {
             !crossSigningPrivateKeys.has("user_signing")
         );
 
-        const keySignatures = {};
+        const keySignatures: Record<string, ISignedKey> = {};
 
         if (selfSigningChanged) {
             logger.info("Got new self-signing key", newCrossSigning.getId("self_signing"));
@@ -1537,7 +1544,7 @@ export class Crypto extends EventEmitter {
             // We may have existing signatures from deleted devices, which will cause
             // the entire upload to fail.
             keySignatures[this.crossSigningInfo.getId()] = Object.assign(
-                {},
+                {} as ISignedKey,
                 masterKey,
                 {
                     signatures: {
@@ -1551,7 +1558,7 @@ export class Crypto extends EventEmitter {
 
         const keysToUpload = Object.keys(keySignatures);
         if (keysToUpload.length) {
-            const upload = ({ shouldEmit }) => {
+            const upload = ({ shouldEmit = false }) => {
                 logger.info(`Starting background key sig upload for ${keysToUpload}`);
                 return this.baseApis.uploadKeySignatures({ [this.userId]: keySignatures })
                     .then((response) => {
@@ -1927,7 +1934,7 @@ export class Crypto extends EventEmitter {
         }
 
         const oneTimeKeys = await this.olmDevice.getOneTimeKeys();
-        const oneTimeJson = {};
+        const oneTimeJson: Record<string, { key: string }> = {};
 
         for (const keyId in oneTimeKeys.curve25519) {
             if (oneTimeKeys.curve25519.hasOwnProperty(keyId)) {
@@ -2076,7 +2083,7 @@ export class Crypto extends EventEmitter {
                 );
                 const device = await this.crossSigningInfo.signUser(xsk);
                 if (device) {
-                    const upload = async ({ shouldEmit }) => {
+                    const upload = async ({ shouldEmit = false }) => {
                         logger.info("Uploading signature for " + userId + "...");
                         const response = await this.baseApis.uploadKeySignatures({
                             [userId]: {
@@ -2149,7 +2156,7 @@ export class Crypto extends EventEmitter {
             logger.info("Own device " + deviceId + " marked verified: signing");
 
             // Signing only needed if other device not already signed
-            let device;
+            let device: ISignedKey;
             const deviceTrust = this.checkDeviceTrust(userId, deviceId);
             if (deviceTrust.isCrossSigningVerified()) {
                 logger.log(`Own device ${deviceId} already cross-signing verified`);
@@ -2160,7 +2167,7 @@ export class Crypto extends EventEmitter {
             }
 
             if (device) {
-                const upload = async ({ shouldEmit }) => {
+                const upload = async ({ shouldEmit = false }) => {
                     logger.info("Uploading signature for " + deviceId);
                     const response = await this.baseApis.uploadKeySignatures({
                         [userId]: {
@@ -2204,11 +2211,7 @@ export class Crypto extends EventEmitter {
             return Promise.resolve(existingRequest);
         }
         const channel = new InRoomChannel(this.baseApis, roomId, userId);
-        return this.requestVerificationWithChannel(
-            userId,
-            channel,
-            this.inRoomVerificationRequests,
-        );
+        return this.requestVerificationWithChannel(userId, channel, this.inRoomVerificationRequests);
     }
 
     public requestVerification(userId: string, devices: string[]): Promise<VerificationRequest> {
@@ -2220,17 +2223,13 @@ export class Crypto extends EventEmitter {
             return Promise.resolve(existingRequest);
         }
         const channel = new ToDeviceChannel(this.baseApis, userId, devices, ToDeviceChannel.makeTransactionId());
-        return this.requestVerificationWithChannel(
-            userId,
-            channel,
-            this.toDeviceVerificationRequests,
-        );
+        return this.requestVerificationWithChannel(userId, channel, this.toDeviceVerificationRequests);
     }
 
     private async requestVerificationWithChannel(
         userId: string,
         channel: IVerificationChannel,
-        requestsMap: any, // TODO types
+        requestsMap: IRequestsMap,
     ): Promise<VerificationRequest> {
         let request = new VerificationRequest(channel, this.verificationMethods, this.baseApis);
         // if transaction id is already known, add request
@@ -2618,7 +2617,7 @@ export class Crypto extends EventEmitter {
         users: string[],
         force?: boolean,
     ): Promise<Record<string, Record<string, olmlib.IOlmSessionResult>>> {
-        const devicesByUser = {};
+        const devicesByUser: Record<string, DeviceInfo[]> = {};
 
         for (let i = 0; i < users.length; ++i) {
             const userId = users[i];
@@ -2651,7 +2650,7 @@ export class Crypto extends EventEmitter {
      * @return {module:crypto/OlmDevice.MegolmSessionData[]} a list of session export objects
      */
     public async exportRoomKeys(): Promise<IMegolmSessionData[]> {
-        const exportedSessions = [];
+        const exportedSessions: IMegolmSessionData[] = [];
         await this.cryptoStore.doTxn(
             'readonly', [IndexedDBCryptoStore.STORE_INBOUND_GROUP_SESSIONS], (txn) => {
                 this.cryptoStore.getAllEndToEndInboundGroupSessions(txn, (s) => {
@@ -2783,8 +2782,7 @@ export class Crypto extends EventEmitter {
             delete content['io.element.performance_metrics'];
         }
 
-        const encryptedContent = await alg.encryptMessage(
-            room, event.getType(), content);
+        const encryptedContent = await alg.encryptMessage(room, event.getType(), content);
 
         if (mRelatesTo) {
             encryptedContent['m.relates_to'] = mRelatesTo;
@@ -3154,7 +3152,7 @@ export class Crypto extends EventEmitter {
         if (!ToDeviceChannel.validateEvent(event, this.baseApis)) {
             return;
         }
-        const createRequest = event => {
+        const createRequest = (event: MatrixEvent) => {
             if (!ToDeviceChannel.canCreateRequest(ToDeviceChannel.getEventType(event))) {
                 return;
             }
@@ -3172,11 +3170,7 @@ export class Crypto extends EventEmitter {
             return new VerificationRequest(
                 channel, this.verificationMethods, this.baseApis);
         };
-        this.handleVerificationEvent(
-            event,
-            this.toDeviceVerificationRequests,
-            createRequest,
-        );
+        this.handleVerificationEvent(event, this.toDeviceVerificationRequests, createRequest);
     }
 
     /**
@@ -3199,7 +3193,7 @@ export class Crypto extends EventEmitter {
         if (!InRoomChannel.validateEvent(event, this.baseApis)) {
             return;
         }
-        const createRequest = event => {
+        const createRequest = (event: MatrixEvent) => {
             const channel = new InRoomChannel(
                 this.baseApis,
                 event.getRoomId(),
@@ -3207,18 +3201,13 @@ export class Crypto extends EventEmitter {
             return new VerificationRequest(
                 channel, this.verificationMethods, this.baseApis);
         };
-        this.handleVerificationEvent(
-            event,
-            this.inRoomVerificationRequests,
-            createRequest,
-            liveEvent,
-        );
+        this.handleVerificationEvent(event, this.inRoomVerificationRequests, createRequest, liveEvent);
     };
 
     private async handleVerificationEvent(
         event: MatrixEvent,
-        requestsMap: any, // TODO types
-        createRequest: any, // TODO types
+        requestsMap: IRequestsMap,
+        createRequest: (event: MatrixEvent) => VerificationRequest,
         isLiveEvent = true,
     ): Promise<void> {
         // Wait for event to get its final ID with pendingEventOrdering: "chronological", since DM channels depend on it.
@@ -3332,7 +3321,7 @@ export class Crypto extends EventEmitter {
                 return;
             }
         }
-        const devicesByUser = {};
+        const devicesByUser: Record<string, DeviceInfo[]> = {};
         devicesByUser[sender] = [device];
         await olmlib.ensureOlmSessionsForDevices(this.olmDevice, this.baseApis, devicesByUser, true);
 
diff --git a/src/crypto/olmlib.ts b/src/crypto/olmlib.ts
index fe90607d9..19c34fe2b 100644
--- a/src/crypto/olmlib.ts
+++ b/src/crypto/olmlib.ts
@@ -28,7 +28,8 @@ import { OlmDevice } from "./OlmDevice";
 import { DeviceInfo } from "./deviceinfo";
 import { logger } from '../logger';
 import { IOneTimeKey } from "./dehydration";
-import { MatrixClient } from "../client";
+import { IClaimOTKsResult, MatrixClient } from "../client";
+import { ISignatures } from "../@types/signed";
 
 enum Algorithm {
     Olm = "m.olm.v1.curve25519-aes-sha2",
@@ -132,6 +133,11 @@ export async function encryptMessageForDevice(
     );
 }
 
+interface IExistingOlmSession {
+    device: DeviceInfo;
+    sessionId?: string;
+}
+
 /**
  * Get the existing olm sessions for the given devices, and the devices that
  * don't have olm sessions.
@@ -152,11 +158,11 @@ export async function getExistingOlmSessions(
     olmDevice: OlmDevice,
     baseApis: MatrixClient,
     devicesByUser: Record<string, DeviceInfo[]>,
-) {
-    const devicesWithoutSession = {};
-    const sessions = {};
+): Promise<[Record<string, DeviceInfo[]>, Record<string, Record<string, IExistingOlmSession>>]> {
+    const devicesWithoutSession: {[userId: string]: DeviceInfo[]} = {};
+    const sessions: {[userId: string]: {[deviceId: string]: IExistingOlmSession}} = {};
 
-    const promises = [];
+    const promises: Promise<void>[] = [];
 
     for (const [userId, devices] of Object.entries(devicesByUser)) {
         for (const deviceInfo of devices) {
@@ -230,10 +236,10 @@ export async function ensureOlmSessionsForDevices(
         force = false;
     }
 
-    const devicesWithoutSession = [
+    const devicesWithoutSession: [string, string][] = [
         // [userId, deviceId], ...
     ];
-    const result = {};
+    const result: {[userId: string]: {[deviceId: string]: IExistingOlmSession}} = {};
     const resolveSession: Record<string, (sessionId?: string) => void> = {};
 
     // Mark all sessions this task intends to update as in progress. It is
@@ -321,9 +327,7 @@ export async function ensureOlmSessionsForDevices(
     let taskDetail = `one-time keys for ${devicesWithoutSession.length} devices`;
     try {
         log.debug(`Claiming ${taskDetail}`);
-        res = await baseApis.claimOneTimeKeys(
-            devicesWithoutSession, oneTimeKeyAlgorithm, otkTimeout,
-        );
+        res = await baseApis.claimOneTimeKeys(devicesWithoutSession, oneTimeKeyAlgorithm, otkTimeout);
         log.debug(`Claimed ${taskDetail}`);
     } catch (e) {
         for (const resolver of Object.values(resolveSession)) {
@@ -337,8 +341,8 @@ export async function ensureOlmSessionsForDevices(
         failedServers.push(...Object.keys(res.failures));
     }
 
-    const otkResult = res.one_time_keys || {};
-    const promises = [];
+    const otkResult = res.one_time_keys || {} as IClaimOTKsResult["one_time_keys"];
+    const promises: Promise<void>[] = [];
     for (const [userId, devices] of Object.entries(devicesByUser)) {
         const userRes = otkResult[userId] || {};
         for (let j = 0; j < devices.length; j++) {
@@ -359,7 +363,7 @@ export async function ensureOlmSessionsForDevices(
             }
 
             const deviceRes = userRes[deviceId] || {};
-            let oneTimeKey = null;
+            let oneTimeKey: IOneTimeKey = null;
             for (const keyId in deviceRes) {
                 if (keyId.indexOf(oneTimeKeyAlgorithm + ":") === 0) {
                     oneTimeKey = deviceRes[keyId];
@@ -441,7 +445,7 @@ async function _verifyKeyAndStartSession(
 
 export interface IObject {
     unsigned?: object;
-    signatures?: object;
+    signatures?: ISignatures;
 }
 
 /**
diff --git a/src/crypto/store/indexeddb-crypto-store-backend.ts b/src/crypto/store/indexeddb-crypto-store-backend.ts
index 3f25a0353..8e311a1a2 100644
--- a/src/crypto/store/indexeddb-crypto-store-backend.ts
+++ b/src/crypto/store/indexeddb-crypto-store-backend.ts
@@ -200,10 +200,10 @@ export class Backend implements CryptoStore {
 
         // index into the wantedStates array
         let stateIndex = 0;
-        let result;
+        let result: OutgoingRoomKeyRequest;
 
-        function onsuccess(ev) {
-            const cursor = ev.target.result;
+        function onsuccess(this: IDBRequest<IDBCursorWithValue>) {
+            const cursor = this.result;
             if (cursor) {
                 // got a match
                 result = cursor.value;
@@ -218,7 +218,7 @@ export class Backend implements CryptoStore {
             }
 
             const wantedState = wantedStates[stateIndex];
-            const cursorReq = ev.target.source.openCursor(wantedState);
+            const cursorReq = (this.source as IDBIndex).openCursor(wantedState);
             cursorReq.onsuccess = onsuccess;
         }
 
@@ -255,10 +255,10 @@ export class Backend implements CryptoStore {
         wantedStates: number[],
     ): Promise<OutgoingRoomKeyRequest[]> {
         let stateIndex = 0;
-        const results = [];
+        const results: OutgoingRoomKeyRequest[] = [];
 
-        function onsuccess(ev) {
-            const cursor = ev.target.result;
+        function onsuccess(this: IDBRequest<IDBCursorWithValue>) {
+            const cursor = this.result;
             if (cursor) {
                 const keyReq = cursor.value;
                 if (keyReq.recipients.includes({ userId, deviceId })) {
@@ -274,7 +274,7 @@ export class Backend implements CryptoStore {
                 }
 
                 const wantedState = wantedStates[stateIndex];
-                const cursorReq = ev.target.source.openCursor(wantedState);
+                const cursorReq = (this.source as IDBIndex).openCursor(wantedState);
                 cursorReq.onsuccess = onsuccess;
             }
         }
@@ -306,10 +306,10 @@ export class Backend implements CryptoStore {
         expectedState: number,
         updates: Partial<OutgoingRoomKeyRequest>,
     ): Promise<OutgoingRoomKeyRequest | null> {
-        let result = null;
+        let result: OutgoingRoomKeyRequest = null;
 
-        function onsuccess(ev) {
-            const cursor = ev.target.result;
+        function onsuccess(this: IDBRequest<IDBCursorWithValue>) {
+            const cursor = this.result;
             if (!cursor) {
                 return;
             }
@@ -444,7 +444,7 @@ export class Backend implements CryptoStore {
         const objectStore = txn.objectStore("sessions");
         const idx = objectStore.index("deviceKey");
         const getReq = idx.openCursor(deviceKey);
-        const results = {};
+        const results: Parameters<Parameters<Backend["getEndToEndSessions"]>[2]>[0] = {};
         getReq.onsuccess = function() {
             const cursor = getReq.result;
             if (cursor) {
@@ -734,7 +734,7 @@ export class Backend implements CryptoStore {
     }
 
     public getEndToEndRooms(txn: IDBTransaction, func: (rooms: Record<string, IRoomEncryption>) => void): void {
-        const rooms = {};
+        const rooms: Parameters<Parameters<Backend["getEndToEndRooms"]>[1]>[0] = {};
         const objectStore = txn.objectStore("rooms");
         const getReq = objectStore.openCursor();
         getReq.onsuccess = function() {
@@ -756,7 +756,7 @@ export class Backend implements CryptoStore {
 
     public getSessionsNeedingBackup(limit: number): Promise<ISession[]> {
         return new Promise((resolve, reject) => {
-            const sessions = [];
+            const sessions: ISession[] = [];
 
             const txn = this.db.transaction(
                 ["sessions_needing_backup", "inbound_group_sessions"],
@@ -877,8 +877,8 @@ export class Backend implements CryptoStore {
         func: (txn: IDBTransaction) => T,
         log: PrefixedLogger = logger,
     ): Promise<T> {
-        let startTime;
-        let description;
+        let startTime: number;
+        let description: string;
         if (PROFILE_TRANSACTIONS) {
             const txnId = this.nextTxnId++;
             startTime = Date.now();
diff --git a/src/crypto/store/localStorage-crypto-store.ts b/src/crypto/store/localStorage-crypto-store.ts
index 08f23affe..bd0b87bc4 100644
--- a/src/crypto/store/localStorage-crypto-store.ts
+++ b/src/crypto/store/localStorage-crypto-store.ts
@@ -175,8 +175,10 @@ export class LocalStorageCryptoStore extends MemoryCryptoStore {
     }
 
     public async filterOutNotifiedErrorDevices(devices: IOlmDevice[]): Promise<IOlmDevice[]> {
-        const notifiedErrorDevices = getJsonItem<string[]>(this.store, KEY_NOTIFIED_ERROR_DEVICES) || {};
-        const ret = [];
+        const notifiedErrorDevices = getJsonItem<MemoryCryptoStore["notifiedErrorDevices"]>(
+            this.store, KEY_NOTIFIED_ERROR_DEVICES,
+        ) || {};
+        const ret: IOlmDevice[] = [];
 
         for (const device of devices) {
             const { userId, deviceInfo } = device;
@@ -291,7 +293,7 @@ export class LocalStorageCryptoStore extends MemoryCryptoStore {
     }
 
     public getEndToEndRooms(txn: unknown, func: (rooms: Record<string, IRoomEncryption>) => void): void {
-        const result = {};
+        const result: Record<string, IRoomEncryption> = {};
         const prefix = keyEndToEndRoomsPrefix('');
 
         for (let i = 0; i < this.store.length; ++i) {
@@ -306,7 +308,7 @@ export class LocalStorageCryptoStore extends MemoryCryptoStore {
 
     public getSessionsNeedingBackup(limit: number): Promise<ISession[]> {
         const sessionsNeedingBackup = getJsonItem<string[]>(this.store, KEY_SESSIONS_NEEDING_BACKUP) || {};
-        const sessions = [];
+        const sessions: ISession[] = [];
 
         for (const session in sessionsNeedingBackup) {
             if (Object.prototype.hasOwnProperty.call(sessionsNeedingBackup, session)) {
diff --git a/src/crypto/verification/request/Channel.ts b/src/crypto/verification/request/Channel.ts
index 88955006b..3bba7d822 100644
--- a/src/crypto/verification/request/Channel.ts
+++ b/src/crypto/verification/request/Channel.ts
@@ -30,4 +30,5 @@ export interface IVerificationChannel {
     sendCompleted(type: string, content: Record<string, any>): Promise<void>;
     completedContentFromEvent(event: MatrixEvent): Record<string, any>;
     canCreateRequest(type: string): boolean;
+    handleEvent(event: MatrixEvent, request: VerificationRequest, isLiveEvent: boolean): Promise<void>;
 }
diff --git a/src/crypto/verification/request/InRoomChannel.ts b/src/crypto/verification/request/InRoomChannel.ts
index f07c39ffb..9e8690e1d 100644
--- a/src/crypto/verification/request/InRoomChannel.ts
+++ b/src/crypto/verification/request/InRoomChannel.ts
@@ -26,6 +26,7 @@ import { IVerificationChannel } from "./Channel";
 import { EventType } from "../../../@types/event";
 import { MatrixClient } from "../../../client";
 import { MatrixEvent } from "../../../models/event";
+import { IRequestsMap } from "../..";
 
 const MESSAGE_TYPE = EventType.RoomMessage;
 const M_REFERENCE = "m.reference";
@@ -304,7 +305,7 @@ export class InRoomChannel implements IVerificationChannel {
     }
 }
 
-export class InRoomRequests {
+export class InRoomRequests implements IRequestsMap {
     private requestsByRoomId = new Map<string, Map<string, VerificationRequest>>();
 
     public getRequest(event: MatrixEvent): VerificationRequest {
@@ -328,7 +329,7 @@ export class InRoomRequests {
         this.doSetRequest(event.getRoomId(), InRoomChannel.getTransactionId(event), request);
     }
 
-    public setRequestByChannel(channel: InRoomChannel, request: VerificationRequest): void {
+    public setRequestByChannel(channel: IVerificationChannel, request: VerificationRequest): void {
         this.doSetRequest(channel.roomId, channel.transactionId, request);
     }
 
diff --git a/src/crypto/verification/request/ToDeviceChannel.ts b/src/crypto/verification/request/ToDeviceChannel.ts
index 538a3db2c..61bd8bc34 100644
--- a/src/crypto/verification/request/ToDeviceChannel.ts
+++ b/src/crypto/verification/request/ToDeviceChannel.ts
@@ -30,8 +30,9 @@ import { errorFromEvent, newUnexpectedMessageError } from "../Error";
 import { MatrixEvent } from "../../../models/event";
 import { IVerificationChannel } from "./Channel";
 import { MatrixClient } from "../../../client";
+import { IRequestsMap } from '../..';
 
-type Request = VerificationRequest<ToDeviceChannel>;
+export type Request = VerificationRequest<ToDeviceChannel>;
 
 /**
  * A key verification channel that sends verification events over to_device messages.
@@ -276,7 +277,7 @@ export class ToDeviceChannel implements IVerificationChannel {
 
     private async sendToDevices(type: string, content: Record<string, any>, devices: string[]): Promise<void> {
         if (devices.length) {
-            const msgMap = {};
+            const msgMap: Record<string, Record<string, any>> = {};
             for (const deviceId of devices) {
                 msgMap[deviceId] = content;
             }
@@ -294,7 +295,7 @@ export class ToDeviceChannel implements IVerificationChannel {
     }
 }
 
-export class ToDeviceRequests {
+export class ToDeviceRequests implements IRequestsMap {
     private requestsByUserId = new Map<string, Map<string, Request>>();
 
     public getRequest(event: MatrixEvent): Request {
diff --git a/src/crypto/verification/request/VerificationRequest.ts b/src/crypto/verification/request/VerificationRequest.ts
index 5d145508b..b6c0d9ef4 100644
--- a/src/crypto/verification/request/VerificationRequest.ts
+++ b/src/crypto/verification/request/VerificationRequest.ts
@@ -582,7 +582,9 @@ export class VerificationRequest<C extends IVerificationChannel = IVerificationC
         // get common methods
         if (phase === PHASE_REQUESTED || phase === PHASE_READY) {
             if (!this.wasSentByOwnDevice(event)) {
-                const content = event.getContent();
+                const content = event.getContent<{
+                    methods: string[];
+                }>();
                 this.commonMethods =
                     content.methods.filter(m => this.verificationMethods.has(m));
             }
diff --git a/src/models/MSC3089TreeSpace.ts b/src/models/MSC3089TreeSpace.ts
index 48209dbc0..d17b4b589 100644
--- a/src/models/MSC3089TreeSpace.ts
+++ b/src/models/MSC3089TreeSpace.ts
@@ -282,7 +282,7 @@ export class MSC3089TreeSpace {
         const members = this.room.currentState.getStateEvents(EventType.RoomMember);
         for (const member of members) {
             const isNotUs = member.getStateKey() !== this.client.getUserId();
-            if (isNotUs && kickMemberships.includes(member.getContent<string>()['membership'])) {
+            if (isNotUs && kickMemberships.includes(member.getContent().membership)) {
                 const stateKey = member.getStateKey();
                 if (!stateKey) {
                     throw new Error("State key not found for branch");