Surprisingly, black holes have a thermodynamic entropy proportional to the area of their horizons. Even more surprisingly, black holes have a fine-grained quantum entropy given by the area of another surface, which typically lies behind the horizon. This characterizes the full quantum entropy of the black hole. Using this notion it has been possible to compute the entropy of Hawking radiation and to obtain a result which is consistent with unitary evolution and is different from Hawking's original result.