  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.
