This talk discusses the application of hybridizable discontinuous Galerkin (HDG) methods to canonical Hamiltonian PDEs. We present necessary and sufficient conditions for an HDG method to satisfy a multisymplectic conservation law, when applied to such a system, and show that these conditions are satisfied by "hybridized" versions of several of the most commonly-used finite element methods. These finite element methods may therefore be used for high-order, structure-preserving discretization of Hamiltonian PDEs on unstructured meshes. (Joint work with Robert McLachlan.)