A mathematical billiard is a system that describes the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of its dynamics is deeply intertwined with the geometric properties of the domain. While it is evident that the shape determines the dynamics, a more subtle and challenging question is how knowledge of the dynamics can be used to reconstruct the shape of the domain. This leads to many intriguing inverse problems and unresolved rigidity questions, which have been the focus of active research in recent decades. In this talk, I will address questions related to the dynamical and spectral rigidity of these systems and discuss recent results obtained toward their resolution.