  This paper gives a survey of wave dynamics in the Kerr space-time geometry,
the mathematical model of a rotating black hole in equilibrium. After a brief
introduction to the Kerr metric, we review the separability properties of
linear wave equations for fields of general spin $s=0, 1/2, 1, 2$,
corresponding to scalar, Dirac, electromagnetic fields and linearized
gravitational waves. We give results on the long-time dynamics of Dirac and
scalar waves, including decay rates for massive Dirac fields. For scalar waves,
we give a rigorous treatment of superradiance and describe rigorously a
mechanism of energy extraction from a rotating black hole. Finally, we discuss
the open problem of linear stability of the Kerr metric and present partial
results.
