  These notes, based on lectures given at the summer school on Asymptotic
Analysis in General Relativity, collect material on the Einstein equations, the
geometry of black hole spacetimes, and the analysis of fields on black hole
backgrounds. The Kerr model of a rotating black hole in vacuum is expected to
be unique and stable. The problem of proving these fundamental facts provides
the background for the material presented in these notes.
  Among the many topics which are relevant for the uniqueness and stability
problems are the theory of fields on black hole spacetimes, in particular for
gravitational perturbations of the Kerr black hole, and more generally, the
study of nonlinear field equations in the presence of trapping. The study of
these questions requires tools from several different fields, including
Lorentzian geometry, hyperbolic differential equations and spin geometry, which
are all relevant to the black hole stability problem.
