Although the evolution of quantum fields in a black-hole background has been considered occasionally, a black hole has always been regarded as a classical entity devoid of quantum dynamics. Such a viewpoint is clearly only an approximation: a black hole must be subject to quantum laws. However, a rigorous quantum discussion of black holes must await the development of a complete quantum theory of gravitation. Nevertheless, it is possible to consider heuristically some quantum features of black holes by means of well-understood principles and arguments from ordinary quantum theory. By this approach we now discuss the possibility that the mass spectrum of the Kerr black hole may be discrete, and we infer a formula for the allowed mass levels. We also consider some applications of this formula…