Black Hole Entropy and Physics at Planckian Scales

We discuss and compare different definitions of the entropy of a black hole. In particular we show that the thermodynamical entropy defined by the response of the free energy of a black hole to the change of temperature does not coincide with the statistical-mechanical entropy, obtained by counting its dynamical degrees of freedom. The no-boundary wavefunction of a black hole, its relation to the entropy problem and its generalization are discussed. We discuss the relation between on-shell and off-shell calculations of the entropy, and the role of the renormalization of the gravitational constant.