Many PDEs arising in physical systems have symmetries and conservation laws that are local in space. However, classical finite element methods are described in terms of spaces of global functions, so it is difficult even to make sense of such local properties. In this talk, I will describe how hybrid finite element methods, based on non-overlapping domain decomposition, provide a way around this local-vs.-global obstacle. Specifically, I will discuss joint work with Robert McLachlan on multisymplectic hybridizable discontinuous Galerkin methods for Hamiltonian PDEs, as well as joint work with Yakov Berchenko-Kogan on symmetry-preserving hybrid finite element methods for gauge theory.[-]