The pioneering work of Bekenstein and Hawking in the 70's produced a universal area law for black hole entropy valid in the infinite size limit. Higher derivative and quantum corrections to the action induce finite size corrections to the black hole entropy. For supersymmetric black holes, one has a notion of exact quantum entropy, which can be compared with the logarithm of the number of microstates of the black hole. I shall explain the basic ideas, and then report on some recent progress in this direction in the context of superstring theory. Time and interest permitting, I shall speak about interesting relations with number theory.