  I describe approaches to the study of black hole spacetimes via numerical
relativity. After a brief review of the basic formalisms and techniques used in
numerical black hole simulations, I discuss a series of calculations from
axisymmetry to full 3D that can be seen as stepping stones to simulations of
the full 3D coalescence of two black holes. In particular, I emphasize the
interplay between perturbation theory and numerical simulation that build both
confidence in present results and tools to aid and to interpret results of
future simulations of black hole coalescence.
