Is black hole evaporation unitary, and if so, what happens to an observer who falls into a black hole? In string theory, the fuzzball program addresses these questions by studying the gravitational descriptions of black hole microstates. In this talk we propose and investigate a classification of supersymmetric D1-D5-P microstates, building on the work of Brown and Henneaux and the AdS/CFT correspondence. This classification enhances the prospect of understanding all microstates of the large
supersymmetric BMPV black hole. We also discuss recent developments in black hole complementarity, and describe a particular realization of complementarity which is conjectured to arise in the fuzzball scenario.