We consider the Dirac and scalar wave equations in the Kerr geometry of a rotating black hole in equilibrium, with Cauchy data in L2, bounded near the event horizon. We derive integral spectral representations for the propagators in terms of the radial and angular decompositions arising from Chandrasekhar’s separation of variables. These are then used to investigate the long term dynamics of Dirac and scalar waves in Kerr geometry. This is joint work with Felix Finster, Joel Smoller and Shing-Tung Yau.