  Understanding the behaviour of linear waves on black hole backgrounds is a
central problem in general relativity, intimately connected with the nonlinear
stability of the black hole spacetimes themselves as solutions to the Einstein
equations--a major open question in the subject. Nonetheless, it is only very
recently that even the most basic boundedness and quantitative decay properties
of linear waves have been proven in a suitably general class of black hole
exterior spacetimes. This talk will review our current mathematical
understanding of waves on black hole backgrounds, beginning with the classical
boundedness theorem of Kay and Wald on exactly Schwarzschild exteriors and
ending with very recent boundedness and decay theorems (proven in collaboration
with Igor Rodnianski) on a wider class of spacetimes. This class of spacetimes
includes in particular slowly rotating Kerr spacetimes, but in the case of the
boundedness theorem is in fact much larger, encompassing general axisymmetric
stationary spacetimes whose geometry is sufficiently close to Schwarzschild and
whose Killing fields span the null generator of the horizon.
