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I shall introduce a simple dynamical system called symplectic billiards. As opposed to the usual (Birkhoff) billiards, where length is the generating function, for symplectic billiards the symplectic area is the generating function. I shall explore basic properties and exhibit several similarities, but also differences, of symplectic billiards to Birkhoff billiards. Symplectic billiards can be defined not only in the plane, but also in linear symplectic spaces. In this multi-dimensional setting, I shall discuss the existence of periodic trajectories and describe the integrable dynamics of symplectic billiards in ellipsoids. Based on joint work with Peter Albers.