I will discuss in some detail the main difficulties of the problem of nonlinear stability of black holes and the recent advances on the related issue of linear stability.

The gravitational waves detected recently by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which, generically, solutions of the initial value problem of the Einstein vacuum equations approach asymptotically, in any compact region, a Kerr black hole. Though the conjecture is so very easy to formulate and happens to be consistent with astrophysical observations as well as numerical experiments, its proof is far beyond our current mathematical understanding, let alone available techniques techniques. In fact even the far simpler and fundamental question of the stability of the Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of new geometric and analytic techniques which I will try to outline in my lectures.