  In this thesis, we consider two different problems relevant to general
relativity. Over the last few years, opinions on physically relevant
singularities occurring in FRW cosmologies have considerably changed. We
present an extensive catalogue of such cosmological milestones using
generalized power series both at the kinematical and dynamical level. We define
the notion of "scale factor singularity" and explore its relation to polynomial
and differential curvature singularities. We also extract dynamical information
using the Friedmann equations and derive necessary and sufficient conditions
for the existence of cosmological milestones such as big bangs, big crunches,
big rips, sudden singularities and extremality events. Specifically, we provide
a complete characterization of cosmological milestones for which the dominant
energy condition is satisfied. The second problem looks at one of the very
small number of serious alternatives to the usual concept of an astrophysical
black hole, that is, the gravastar model developed by Mazur and Mottola. By
considering a generalized class of similar models with continuous pressure (no
infinitesimally thin shells) and negative central pressure, we demonstrate that
gravastars cannot be perfect fluid spheres: anisotropic pressures are
unavoidable. We provide bounds on the necessary anisotropic pressure and show
that these transverse stresses that support a gravastar permit a higher
compactness than is given by the Buchdahl-Bondi bound for perfect fluid stars.
We also comment on the qualitative features of the equation of state that such
gravastar-like objects without any horizon must have.
