  A deeper understanding of the thermal properties of black holes than we
presently have depends to a large degree on obtaining a firmer grasp of the
properties of the entropy. For such an understanding we must at least know the
basic relations among entropy, energy, and temperature of a black hole in
thermal equilibrium with quantized matter fields. Limiting attention to
spherical, uncharged (``Schwarzschild") holes, we will find that the basic
Bekenstein-Hawking relations have to be generalized when the hole is dressed by
quantum fields. Though this fact is not surprising, the corrections contain
surprises and are very instructive. My purpose here is to discuss several
aspects of this problem and to display some concrete results within the
framework of the semi-classical theory of quantum fields in curved spacetime.
