Extremal black holes are special types of black holes which have exactly zero temperature. I will present a proof that extremal black holes form in finite time in gravitational collapse of charged matter. In particular, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. I will also present a result which shows that extremal black holes take on a central role in gravitational collapse, giving rise to a new conjectural picture of “extremal critical collapse.” This is joint work with Ryan Unger (Princeton).