  In this review we consider first order gravity in four dimensions. In
particular, we focus our attention in formulations where the fundamental
variables are a tetrad $e_a^I$ and a SO(3,1) connection ${\omega_{aI}}^J$. We
study the most general action principle compatible with diffeomorphism
invariance. This implies, in particular, considering besides the standard
Einstein-Hilbert-Palatini term, other terms that either do not change the
equations of motion, or are topological in nature. Having a well defined action
principle sometimes involves the need for additional boundary terms, whose
detailed form may depend on the particular boundary conditions at hand. In this
work, we consider spacetimes that include a boundary at infinity, satisfying
asymptotically flat boundary conditions and/or an internal boundary satisfying
isolated horizons boundary conditions. We focus on the covariant Hamiltonian
formalism where the phase space $\Gamma$ is given by solutions to the equations
of motion. For each of the possible terms contributing to the action we
consider the well posedness of the action, its finiteness, the contribution to
the symplectic structure, and the Hamiltonian and Noether charges. For the
chosen boundary conditions, standard boundary terms warrant a well posed
theory. Furthermore, the boundary and topological terms do not contribute to
the symplectic structure, nor the Hamiltonian conserved charges. The Noether
conserved charges, on the other hand, do depend on such additional terms. The
aim of this manuscript is to present a comprehensive and self-contained
treatment of the subject, so the style is somewhat pedagogical. Furthermore,
along the way we point out and clarify some issues that have not been clearly
understood in the literature.
