  We present a numerical study of general relativistic boson stars in both
spherical symmetry and axisymmetry. We consider both time-independent problems,
involving the solution of equilibrium equations for rotating boson stars, and
time-dependent problems, focusing on black hole critical behaviour associated
with boson stars.
  The study of the critical phenomena that arise at the threshold of black hole
formation has been a subject of intense interest among relativists and applied
mathematicians over the past decade. Type I critical phenomena were previously
observed in the dynamics of spherically symmetric boson stars by Hawley and
Choptuik. We extend this work and show that, contrary to previous claims, the
subcritical end-state is well described by a stable boson star executing a
large amplitude oscillation with a frequency in good agreement with that
predicted for the fundamental normal mode of the end-state star from linear
perturbation theory.
  We then extend our studies of critical phenomena to the axisymmetric case,
studying two distinct classes of parametrized families of initial data whose
evolution generates families of spacetimes that ``interpolate'' between those
than contain a black hole and those that do not. In both cases we find strong
evidence for a Type I transition at threshold, and are able to demonstrate
scaling of the lifetime for near-critical configurations of the type expected
for such a transition. This is the first time that Type I critical solutions
have been simulated in axisymmetry.
