What is the spectrum of the Laplacian operator in a domain?

This problem is known as "hearing the shape of a drum". Is it possible to know the shape of a drum (a 2d membrane with a fixed boundary) only by hearing the sound it produces (its spectrum) ?

As nice as the idea sounds, the answer is negative. Gordon, Webb and Wolpert gave a counterexample, with two domains ("drums") that have the same spectrum under the laplacian operator:

(image from https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum)

However, the answer can be made positive under some additional restrictions on the domain, such as only considering convex domains with well-behaved boundaries. We don't know how much these restrictions can be relaxed! It's an interesting problem.