Extreme black holes are the subset of stationary black holes characterized by a degenerate Killing horizon. They allow one to consider the black hole classification problem in a constrained setting and occupy a privileged position in various geometric and physical contexts. In particular, supersymmetric black holes (those saturating a BPS bound) are necessarily extreme. I will discuss how there is a precise sense in which spacetime near the event horizon of an axisymmetric extreme black hole has the geometry of a fibration over AdS_2 or AdS_3. I will also discuss `decoupling limits' of recently-found extreme 5d black holes that give rise to spacetimes that interpolate between different AdS spaces.