In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. In the talk I will describe the geometry of these structures, define pseudo-Euclidean billiards and discuss their properties. In particular, I will outline integrability of the null billiard in the ellipsoid and the null geodesic flow on the ellipsoid in pseudo-Euclidean space, that is a counterpart of the classical Poncelet theorem and an example of a contact completely integrable system.