Appears in collection : Balzan Conference on Open Issues in Gravitation

The third law of black hole mechanics asserts that it is impossible for a non-extremal black hole to become extremal in finite time (in classical General Relativity). Counterexamples were found recently: gravitational collapse of a massless charged scalar field can produce an exactly extremal Reissner-Nordstrom black hole in finite time, passing through an intermediate phase in which the solution is exactly Schwarzschild at the horizon. These examples involve matter with a large charge to mass ratio. I will describe how if the charge to mass ratio of matter is suitably bounded then one cannot form an extremal Reissner-Nordstrom black hole in finite time. It is conjectured that one can form an extremal rotating black hole via gravitational collapse of gravitational waves. I will describe recent work showing that this conjecture is true in five spacetime dimensions.