My talk is about the well-known ``inverse spectral problem'' for convex bounded smooth plane domains, i.e. the Kac problem `can you hear the shape of a drum'. The problem is to determine the domain from the eigenvalues of the Laplacian with Dirichlet (or Neumann) boundary conditions. It has long been conjectured that ellipses are uniquely determined by their eigenvalues. My talk is about a recent result with Hamid Hezari which proves that this is true if the ellipse has small eccentricity.