  We provide a rigorous definition of quasi-normal modes for a rotating black
hole. They are given by the poles of a certain meromorphic family of operators
and agree with the heuristic definition in the physics literature. If the black
hole rotates slowly enough, we show that these poles form a discrete subset of
the complex plane. As an application we prove that the local energy of linear
waves in that background decays exponentially once orthogonality to the zero
resonance is imposed.
