Exact quantum entropy of black holes 1. From Bekenstein-Hawking to AdS_2 functional integrals

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.  I shall describe a framework to take these effects into account in a systematic manner for extremal black holes .  I shall then report on progress in the computation of the \emph{exact} quantum entropy of supersymmetric black holes in a supersymmetric theory of gravity. This involves computing a functional integral on $AdS_2$ pace.  I will end by sketching the basic ideas in the computation of the quantum entropy for black holes in superstring theory.