  This thesis is divided in two parts, each one addressing problems that can be
relevant in the study of compact objects. The first part deals with the study
of a magnetized and self-gravitating gas of degenerated fermions (electrons and
neutrons) as sources of a Bianchi-I space-time. We solve numerically the
Einstein-Maxwell field equations for a large set of initial conditions of the
dynamical variables. The collapsing singularity is isotropic for the neutron
gas and can be anisotropic for the electron gas. This result is consistent with
the fact that electrons exhibit a stronger coupling with the magnetic field,
which is the source of anisotropy in the dynamical variables. In the second
part we calculate the entropy of extremal black holes in 4 and 5 dimensions,
using the entropy function formalism of Sen and taking into account higher
order derivative terms that come from the complete set of Riemann invariants.
The resulting entropies show the deviations from the well know
Bekenstein-Hawking area law.
