diff --git a/babel.config.js b/babel.config.cjs
similarity index 100%
rename from babel.config.js
rename to babel.config.cjs
diff --git a/index-wasm-esm.mjs b/index-wasm-esm.js
similarity index 100%
rename from index-wasm-esm.mjs
rename to index-wasm-esm.js
diff --git a/index.mjs b/index.cjs
similarity index 68%
rename from index.mjs
rename to index.cjs
index b792f114f..9b27f4e02 100644
--- a/index.mjs
+++ b/index.cjs
@@ -15,21 +15,16 @@
 // @ts-check
 
 /**
- * This is the entrypoint on non-node ESM environments.
+ * This is the entrypoint on non-node CommonJS environments.
  * `asyncLoad` will load the WASM module using a `fetch` call.
  */
 
-import * as bindings from "./pkg/matrix_sdk_crypto_wasm_bg.js";
+const bindings = require("./pkg/matrix_sdk_crypto_wasm_bg.cjs");
 
-const moduleUrl = new URL("./pkg/matrix_sdk_crypto_wasm_bg.wasm", import.meta.url);
+const moduleUrl = require.resolve("./pkg/matrix_sdk_crypto_wasm_bg.wasm");
 
-// Although we could simply instantiate the WASM at import time with a top-level `await`,
-// we avoid that, to make it easier for callers to delay loading the WASM (and instead
-// wait until `initAsync` is called). (Also, Safari 14 doesn't support top-level `await`.)
-//
-// However, having done so, there is no way to synchronously load the WASM if the user ends
-// up using the bindings before calling `initAsync` (unlike under Node.js), so we just throw
-// an error.
+// We want to throw an error if the user tries to use the bindings before
+// calling `initAsync`.
 bindings.__wbg_set_wasm(
     new Proxy(
         {},
@@ -44,7 +39,7 @@ bindings.__wbg_set_wasm(
 );
 
 /**
- * Stores a promise of the `loadModuleAsync` call
+ * Stores a promise of the `loadModule` call
  * @type {Promise<void> | null}
  */
 let modPromise = null;
@@ -72,10 +67,13 @@ async function loadModuleAsync() {
  *
  * @returns {Promise<void>}
  */
-export async function initAsync() {
+async function initAsync() {
     if (!modPromise) modPromise = loadModuleAsync();
     await modPromise;
 }
 
-// Re-export everything from the generated javascript wrappers
-export * from "./pkg/matrix_sdk_crypto_wasm_bg.js";
+module.exports = {
+    // Re-export everything from the generated javascript wrappers
+    ...bindings,
+    initAsync,
+};
diff --git a/index.js b/index.js
index 9b27f4e02..b792f114f 100644
--- a/index.js
+++ b/index.js
@@ -15,16 +15,21 @@
 // @ts-check
 
 /**
- * This is the entrypoint on non-node CommonJS environments.
+ * This is the entrypoint on non-node ESM environments.
  * `asyncLoad` will load the WASM module using a `fetch` call.
  */
 
-const bindings = require("./pkg/matrix_sdk_crypto_wasm_bg.cjs");
+import * as bindings from "./pkg/matrix_sdk_crypto_wasm_bg.js";
 
-const moduleUrl = require.resolve("./pkg/matrix_sdk_crypto_wasm_bg.wasm");
+const moduleUrl = new URL("./pkg/matrix_sdk_crypto_wasm_bg.wasm", import.meta.url);
 
-// We want to throw an error if the user tries to use the bindings before
-// calling `initAsync`.
+// Although we could simply instantiate the WASM at import time with a top-level `await`,
+// we avoid that, to make it easier for callers to delay loading the WASM (and instead
+// wait until `initAsync` is called). (Also, Safari 14 doesn't support top-level `await`.)
+//
+// However, having done so, there is no way to synchronously load the WASM if the user ends
+// up using the bindings before calling `initAsync` (unlike under Node.js), so we just throw
+// an error.
 bindings.__wbg_set_wasm(
     new Proxy(
         {},
@@ -39,7 +44,7 @@ bindings.__wbg_set_wasm(
 );
 
 /**
- * Stores a promise of the `loadModule` call
+ * Stores a promise of the `loadModuleAsync` call
  * @type {Promise<void> | null}
  */
 let modPromise = null;
@@ -67,13 +72,10 @@ async function loadModuleAsync() {
  *
  * @returns {Promise<void>}
  */
-async function initAsync() {
+export async function initAsync() {
     if (!modPromise) modPromise = loadModuleAsync();
     await modPromise;
 }
 
-module.exports = {
-    // Re-export everything from the generated javascript wrappers
-    ...bindings,
-    initAsync,
-};
+// Re-export everything from the generated javascript wrappers
+export * from "./pkg/matrix_sdk_crypto_wasm_bg.js";
diff --git a/node.mjs b/node.cjs
similarity index 81%
rename from node.mjs
rename to node.cjs
index 65c9cca4d..190b4fd2e 100644
--- a/node.mjs
+++ b/node.cjs
@@ -15,16 +15,16 @@
 // @ts-check
 
 /**
- * This is the entrypoint on node-compatible ESM environments.
+ * This is the entrypoint on node-compatible CommonJS environments.
  * `asyncLoad` will use `fs.readFile` to load the WASM module.
  */
 
-import { fileURLToPath } from "node:url";
-import { readFileSync } from "node:fs";
-import { readFile } from "node:fs/promises";
-import * as bindings from "./pkg/matrix_sdk_crypto_wasm_bg.js";
+const { readFileSync } = require("node:fs");
+const { readFile } = require("node:fs/promises");
+const path = require("node:path");
+const bindings = require("./pkg/matrix_sdk_crypto_wasm_bg.cjs");
 
-const filename = fileURLToPath(new URL("./pkg/matrix_sdk_crypto_wasm_bg.wasm", import.meta.url));
+const filename = path.join(__dirname, "pkg/matrix_sdk_crypto_wasm_bg.wasm");
 
 // In node environments, we want to automatically load the WASM module
 // synchronously if the consumer did not call `initAsync`. To do so, we install
@@ -59,7 +59,7 @@ let initialised = false;
  *
  * It will throw if there is an attempt to load the module asynchronously running
  *
- * @returns {typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d.ts")}
+ * @returns {typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d")}
  */
 function loadModuleSync() {
     if (modPromise) throw new Error("The WASM module is being loaded asynchronously but hasn't finished");
@@ -73,14 +73,14 @@ function loadModuleSync() {
 
     initInstance(instance);
 
-    // @ts-expect-error: Typescript doesn't know what the module exports are
+    // @ts-expect-error: Typescript doesn't know what the instance exports exactly
     return instance.exports;
 }
 
 /**
  * Loads and instantiates the WASM module asynchronously
  *
- * @returns {Promise<typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d.ts")>}
+ * @returns {Promise<typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d")>}
  */
 async function loadModuleAsync() {
     const bytes = await readFile(filename);
@@ -91,7 +91,7 @@ async function loadModuleAsync() {
 
     initInstance(instance);
 
-    // @ts-expect-error: Typescript doesn't know what the module exports are
+    // @ts-expect-error: Typescript doesn't know what the instance exports exactly
     return instance.exports;
 }
 
@@ -103,7 +103,7 @@ async function loadModuleAsync() {
 function initInstance(instance) {
     if (initialised) throw new Error("initInstance called twice");
     bindings.__wbg_set_wasm(instance.exports);
-    // @ts-expect-error: Typescript doesn't know what the module exports are
+    // @ts-expect-error: Typescript doesn't know what the instance exports exactly
     instance.exports.__wbindgen_start();
     initialised = true;
 }
@@ -115,11 +115,14 @@ function initInstance(instance) {
  *
  * @returns {Promise<void>}
  */
-export async function initAsync() {
+async function initAsync() {
     if (initialised) return;
     if (!modPromise) modPromise = loadModuleAsync();
     await modPromise;
 }
 
-// Re-export everything from the generated javascript wrappers
-export * from "./pkg/matrix_sdk_crypto_wasm_bg.js";
+module.exports = {
+    // Re-export everything from the generated javascript wrappers
+    ...bindings,
+    initAsync,
+};
diff --git a/node.js b/node.js
index 190b4fd2e..65c9cca4d 100644
--- a/node.js
+++ b/node.js
@@ -15,16 +15,16 @@
 // @ts-check
 
 /**
- * This is the entrypoint on node-compatible CommonJS environments.
+ * This is the entrypoint on node-compatible ESM environments.
  * `asyncLoad` will use `fs.readFile` to load the WASM module.
  */
 
-const { readFileSync } = require("node:fs");
-const { readFile } = require("node:fs/promises");
-const path = require("node:path");
-const bindings = require("./pkg/matrix_sdk_crypto_wasm_bg.cjs");
+import { fileURLToPath } from "node:url";
+import { readFileSync } from "node:fs";
+import { readFile } from "node:fs/promises";
+import * as bindings from "./pkg/matrix_sdk_crypto_wasm_bg.js";
 
-const filename = path.join(__dirname, "pkg/matrix_sdk_crypto_wasm_bg.wasm");
+const filename = fileURLToPath(new URL("./pkg/matrix_sdk_crypto_wasm_bg.wasm", import.meta.url));
 
 // In node environments, we want to automatically load the WASM module
 // synchronously if the consumer did not call `initAsync`. To do so, we install
@@ -59,7 +59,7 @@ let initialised = false;
  *
  * It will throw if there is an attempt to load the module asynchronously running
  *
- * @returns {typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d")}
+ * @returns {typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d.ts")}
  */
 function loadModuleSync() {
     if (modPromise) throw new Error("The WASM module is being loaded asynchronously but hasn't finished");
@@ -73,14 +73,14 @@ function loadModuleSync() {
 
     initInstance(instance);
 
-    // @ts-expect-error: Typescript doesn't know what the instance exports exactly
+    // @ts-expect-error: Typescript doesn't know what the module exports are
     return instance.exports;
 }
 
 /**
  * Loads and instantiates the WASM module asynchronously
  *
- * @returns {Promise<typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d")>}
+ * @returns {Promise<typeof import("./pkg/matrix_sdk_crypto_wasm_bg.wasm.d.ts")>}
  */
 async function loadModuleAsync() {
     const bytes = await readFile(filename);
@@ -91,7 +91,7 @@ async function loadModuleAsync() {
 
     initInstance(instance);
 
-    // @ts-expect-error: Typescript doesn't know what the instance exports exactly
+    // @ts-expect-error: Typescript doesn't know what the module exports are
     return instance.exports;
 }
 
@@ -103,7 +103,7 @@ async function loadModuleAsync() {
 function initInstance(instance) {
     if (initialised) throw new Error("initInstance called twice");
     bindings.__wbg_set_wasm(instance.exports);
-    // @ts-expect-error: Typescript doesn't know what the instance exports exactly
+    // @ts-expect-error: Typescript doesn't know what the module exports are
     instance.exports.__wbindgen_start();
     initialised = true;
 }
@@ -115,14 +115,11 @@ function initInstance(instance) {
  *
  * @returns {Promise<void>}
  */
-async function initAsync() {
+export async function initAsync() {
     if (initialised) return;
     if (!modPromise) modPromise = loadModuleAsync();
     await modPromise;
 }
 
-module.exports = {
-    // Re-export everything from the generated javascript wrappers
-    ...bindings,
-    initAsync,
-};
+// Re-export everything from the generated javascript wrappers
+export * from "./pkg/matrix_sdk_crypto_wasm_bg.js";
diff --git a/package.json b/package.json
index cbd7a27a0..4ce4ddaa1 100644
--- a/package.json
+++ b/package.json
@@ -18,21 +18,22 @@
         "ruma",
         "nio"
     ],
+    "type": "module",
     "exports": {
         ".": {
             "matrix-org:wasm-esm": {
                 "types": "./index.d.ts",
-                "default": "./index-wasm-esm.mjs"
+                "default": "./index-wasm-esm.js"
             },
             "require": {
                 "types": "./index.d.ts",
-                "node": "./node.js",
-                "default": "./index.js"
+                "node": "./node.cjs",
+                "default": "./index.cjs"
             },
             "import": {
                 "types": "./index.d.ts",
-                "node": "./node.mjs",
-                "default": "./index.mjs"
+                "node": "./node.js",
+                "default": "./index.js"
             }
         }
     },