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.

We explain how inequalities similar to the Penrose inequality provide a simple way of demonstrating classical instability of certain black hole solutions.

The decade 1963-1973 is sometimes referred to as the 'golden age' of black hole research. Mathematical theorems proved in this period led to a complete classical description of equilibrium black holes in four space-time dimensions. More recently, string theory has forced us to take seriously the possibility of extra spatial dimensions. Thinking about black holes in more than four dimensions has led to spectacular advances in our understanding of quantum gravity. In the last five years, we have learned that higher-dimensional black holes exhibit much richer behaviour than four-dimensional ones, even classically. I shall review the discoveries of this new 'silver age' of black hole research. The talk should be accessible to MSc students.