Inverse spectral geometry asks the extent to which geometric and topological information is encoded in spectral data. For geometric objects, we will primarily consider Riemannian surfaces, especially bounded Euclidean domains. We will look at two types of spectral data: the spectrum of the Laplacian (with Dirichlet or Neumann boundary conditions) and the Steklov spectrum, focusing primarily on the latter. The inverse spectral problem for the Laplacian is sometimes phrased as ``Can one hear the shape of a drum?'' since the eigenvalues of a plane domain correspond to the characteristic frequencies of vibration of the domain viewed as a vibrating membrane. The Steklov spectrum of a bounded domain or of a Riemannian manifold M with boundary is the eigenvalue spectrum of the so-called Dirichlet-to-Neumann operator, which inputs smooth functions u on the boundary of M and outputs the normal derivative across the boundary of the unique harmonic extension of u to M. The study of the Steklov spectrum, first introduced in 1902, has seen a surge of interest in recent decades with striking results. After introducing the spectra, we will give a sampling of techniques, progress and open questions.
