A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain (e.g. the shape of the billiard table). While it is evident how the shape determines the dynamics, a more subtle and difficult question is to which extent the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing unanswered questions and difficult conjectures that have been the focus of active research over the last decades. In this talk, we shall describe several of these questions, with particular emphasis on results obtained related to the classification of integrable billiards (also known as Birkhoff conjecture) and to the possibility of inferring dynamical information on the billiard map from its Length Spectrum (i.e., the lengths of its periodic orbits).