Decoupled, separable equations describing perturbations of a Kerr black hole are derived. These equations can be used to study black-hole processes involving scalar, electromagnetic, neutrino or gravitational fields. A number of astrophysical applications are made: Misner's idea that gravitational synchrotron radiation might explain Weber's observations is shown to be untenable; rotating black holes are shown to be stable against small perturbations; energy amplification by "superradiant scattering" of waves off a rotating black hole is computed; the "spin down" (loss of angular momentum) of a rotating black hole caused by a stationary non-axisymmetric perturbation is calculated.
