A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Let $G$ a compact group of isometries of a closed riemannian manifold $(X,m)$. Sunada proved that if $\Gamma_1$ and $\Gamma_2$ are two finite almost-conjugated subgroups of $G$, then $\Gamma_1\setminus X,m)$ and $\Gamma_2\setminus X,m)$ are isospectral. We prove that if $G$ is finite, there exists an open dense set in the set of $G$-invariant metrics for which the converse of this resukt is true. If $G$ is infinite, the situations is more complicated and we obtain some partial results.