  It is well known that both the symplectic structure and the Poisson brackets
of classical field theory can be constructed directly from the Lagrangian in a
covariant way, without passing through the non-covariant canonical Hamiltonian
formalism. This is true even in the presence of constraints and gauge
symmetries. These constructions go under the names of the covariant phase space
formalism and the Peierls bracket. We review both of them, paying more careful
attention, than usual, to the precise mathematical hypotheses that they
require, illustrating them in examples. Also an extensive historical overview
of the development of these constructions is provided. The novel aspect of our
presentation is a significant expansion and generalization of an elegant and
quite recent argument by Forger & Romero showing the equivalence between the
resulting symplectic and Poisson structures without passing through the
canonical Hamiltonian formalism as an intermediary. We generalize it to cover
theories with constraints and gauge symmetries and formulate precise sufficient
conditions under which the argument holds. These conditions include a local
condition on the equations of motion that we call hyperbolizability, and some
global conditions of cohomological nature. The details of our presentation may
shed some light on subtle questions related to the Poisson structure of gauge
theories and their quantization.
