A mathematical billiard is a system describing the inertial motion of
a point mass inside a domain, with elastic reflections at the
boundary. This simple model has been first proposed by G.D. Birkhoff
as a mathematical playground where "it the formal side, usually so
formidable in dynamics, almost completely disappears and only the
interesting qualitative questions need to be considered".
Since then billiards have captured much attention in many different
contexts, becoming a very popular subject of investigation. Despite
their apparently simple (local) dynamics, their qualitative dynamical
properties are extremely non-local. This global influence on the
dynamics translates into several intriguing rigidity phenomena, which
are at the basis of several unanswered questions and conjectures.
In this talk I shall focus on some of these questions. In particular,
I shall describe some recent results related to the classification of
integrable billiards (also known as Birkhoff conjecture).