There are two different standard ways of endowing a physical theory with
a symplectic structure: the canonical and the covariant. The former is
derived from the well-known symplectic structure of a certain cotangent
bundle. The latter is based on the variational calculus. Including a
boundary in the canonical formalism poses no problem; however, in the
covariant formalism, things break apart. In the first part of the
seminar, I will briefly introduce both formalisms without boundary with
a few examples and show their equivalence. The second part of the talk
will be devoted to explaining in detail a new framework that allows us
to include boundaries in a straightforward way in the covariant phase
space formalism. I will illustrate this formalism with several gravity
theories.