Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar domains which are essentially different, but share the same spectrum. The known isospectral domains are not convex, and I think that there are no known isospectral convex domains.

Still, there are a few cases in which the spectrum determines (up to a rigid motion) the shape. For instance we have the circles and rectangles. 

> Are there other classes for which we know that the spectrum completely determines the shape?

> Are there any results on "almost isospectral convex domains"? (i.e. the fact that $\lambda_k(\Omega_1)=\lambda_k(\Omega_2)$ for $k \leq n$ can say something about the resemblance of the convex sets $\Omega_1$ and $\Omega_2$)

If you know any good references on these matters, please share them (especially survey articles).