  The crucial role played by pressure in general relativity is explored in the
mathematically simple context of a static spherically symmetric geometry. By
keeping all pressure terms, the standard formalisms of rotation curve and
gravitational lensing observations are extended to a first post-Newtonian
order. It turns out that both post-Newtonian formalisms encode the
gravitational field differently. Therefore, combined observations of rotation
curves and gravitational lensing of the same galaxy can in principle be used to
infer both the density and pressure profile of the galactic fluid.
  The newly introduced post-Newtonian formalism might allow us to make
inferences about the equation of state of dark matter. While the Cold Dark
Matter paradigm is currently favoured in the astrophysics and cosmology
communities, the formalism presented herein offers an unprecedented way of
being able to directly observe the equation of state.
  In a logically distinct analysis, I investigate the effects of negative
pressure in compact objects, motivated by the recently introduced gravastar
model. I find that gravastar like objects can in principle mimic the external
gravitational field of a black hole. Unlike a black hole, however, gravastars
neither exhibit a pathological curvature singularity at the origin nor do they
posess an event horizon. Instead they are mathematically well defined
everywhere. Finally, another exotic option is considered as a mathematical
alternative to black holes: The anti-gravastar, which is characterized by a
core that has a negative mass-energy density.
