In the previous talk, I have left out the so-called extremal or maximally rotating case of Kerr black holes, i.e. black holes with parameters |a|=M, which in a famous analogy with thermodynamics put forth in the early 1970s, correspond to black holes of vanishing temperature. Extremal black holes have long been a subject of special fascination, both from the astrophysical and the high energy physics point of view. Discussion of them in the literature has been largely dominated by two paradigms, what one could call the “third law paradigm”, which viewed extremality as an unattainable limit, and the “overspinning/overcharging paradigm”, which viewed extremality as a harbinger of so-called “naked singularities”. I will describe the history of both paradigms and explain why we now believe them to be false. I will then discuss a new conjectured picture for the moduli space of vacuum spacetimes in a neighbourhood of extremal Kerr black holes, and the challenges for a potential proof (or disproof!) of this picture.