We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws related to Poincaré invariance within the different approaches. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.