Within the gauge/gravity correspondence, black holes are dual to CFT thermal states. I will discuss asymptotically global AdS_5 × S^5 black holes that are localised on the S^5. These are solutions of type IIB supergravity with S^8 horizon topology that dominate the theory in the microcanonical ensemble at small energies. At higher energies, there is a first-order phase transition to AdS_5-Schwarzschild×S^5. By the AdS/CFT correspondence, this transition is dual to spontaneously breaking the SO(6) R-symmetry of N = 4 super Yang-Mills down to SO(5). I will also briefly discuss localised and nonuniform thermal states of super-Yang-Mills on a circle at strong coupling and associated phase transitions that are reproduced by lattice computations. Slides can be found here.