  These lecture notes, based on a course given at the Zurich Clay Summer School
(June 23-July 18, 2008), review our current mathematical understanding of the
global behaviour of waves on black hole exterior backgrounds. Interest in this
problem stems from its relationship to the non-linear stability of the black
hole spacetimes themselves as solutions to the Einstein equations, one of the
central open problems of general relativity. After an introductory discussion
of the Schwarzschild geometry and the black hole concept, the classical theorem
of Kay and Wald on the boundedness of scalar waves on the exterior region of
Schwarzschild is reviewed. The original proof is presented, followed by a new
more robust proof of a stronger boundedness statement. The problem of decay of
scalar waves on Schwarzschild is then addressed, and a theorem proving
quantitative decay is stated and its proof sketched. This decay statement is
carefully contrasted with the type of statements derived heuristically in the
physics literature for the asymptotic tails of individual spherical harmonics.
Following this, our recent proof of the boundedness of solutions to the wave
equation on axisymmetric stationary backgrounds (including slowly-rotating Kerr
and Kerr-Newman) is reviewed and a new decay result for slowly-rotating Kerr
spacetimes is stated and proved. This last result was announced at the summer
school and appears in print here for the first time. A discussion of the
analogue of these problems for spacetimes with a positive cosmological constant
follows. Finally, a general framework is given for capturing the red-shift
effect for non-extremal black holes. This unifies and extends some of the
analysis of the previous sections. The notes end with a collection of open
problems.
