Can you write async tests that expect toThrow?

You can test your async function like this:

it('should test async errors', async () =>  {        
    await expect(failingAsyncTest())
    .rejects
    .toThrow('I should fail');
});

'I should fail' string will match any part of the error thrown.

Custom Error Class

The use of rejects.toThrow will not work for you. Instead, you can combine the rejects method with the toBeInstanceOf matcher to match the custom error that has been thrown.

Example

it("should test async errors", async () => {
  await expect(asyncFunctionWithCustomError()).rejects.toBeInstanceOf(
    CustomError
  )
})

I'd like to just add on to this and say that the function you're testing must throw an actual Error object throw new Error(...). Jest does not seem to recognize if you just throw an expression like throw 'An error occurred!'.

await expect(async () => { 
    await someAsyncFunction(someParams); 
}).rejects.toThrowError("Some error message");

We must wrap the code in a function to catch the error. Here we are expecting the Error message thrown from someAsyncFunction should be equal to "Some error message". We can call the exception handler also

await expect(async () => { 
    await someAsyncFunction(someParams); 
}).rejects.toThrowError(new InvalidArgumentError("Some error message"));

Read more https://jestjs.io/docs/expect#tothrowerror