Best Way to Test Promises in Jest

Nowadays you can write it in this way as well: docs

describe('Fetching', () => {
    const filters = {
        startDate: '2015-09-01'
    };
    const api = new TestApiTransport(); 

 it('should reject if no startdate is given', () => {
   expect.assertions(1);
   return expect(MyService.fetch()).rejects.toEqual({
     error: 'Your code message',
   });
 });          


 it('should return expected data', () => {
   expect.assertions(1);
   return expect(MyService.fetch(filters, null, api)).resolves.toEqual(extectedObjectFromApi);
 });            
});

Update (06.01.2019)

Agree that the accepted answer doesn’t work correctly as line
expect.assertions(1); does all the magic. Link to docs

expect.assertions(number) verifies that a certain number of assertions
are called during a test. This is often useful when testing
asynchronous code, in order to make sure that assertions in a callback
actually got called.

So putting this line at the top will control that the specific number of assertions are made by the time when the test is run.