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In this talk I will outline a description of the reduced phase space of gravity, for a space or time-like boundary, with and without gauge and matter fields. The reduced phase space is the space of possible initial conditions and will be presented as the reduction of a symplectic space with respect to a coisotropic submanifold (first class constraints) using the Kijowski and Tulczijew construction. Furthermore, if time permits, I will explain how to describe the reduced phase space cohomologically using the BFV formalism.