A microstate geometry is a smooth, time-independent, asymptotically flat, horizon-free solution of type IIB supergravity. According to the “fuzzball" conjecture, such solutions describe individual microstates of black holes. Non-supersymmetric microstate geometries typically suffer from linearized instabilities. I will argue that supersymmetric microstate geometries suffer from a nonlinear instability. I will also discuss how such solutions lead to a new type of mathematical structure, so-called “ambipolar” hyperkahler manifolds, and explain how such manifolds can be constructed.