We establish a connection between Minkowski billiards and symplectic billiards, two classes of dynamical systems that have been studied largely independently. We show that the Minkowski billiard map can be described in symplectic terms via reduction from the canonical symplectic structure on V \times V^*, and that symplectic billiards can be viewed as a ``square root'' of a symplectic version of Minkowski billiards. Joint work with Ana Chavez Caliz and Serge Tabachnikov.