  The macroscopic entropy and the attractor equations for BPS black holes in
four-dimensional N=2 supergravity theories follow from a variational principle
for a certain `entropy function'. We present this function in the presence of
R^2-interactions and non-holomorphic corrections. The variational principle
identifies the entropy as a Legendre transform and this motivates the
definition of various partition functions corresponding to different ensembles
and a hierarchy of corresponding duality invariant inverse Laplace integral
representations for the microscopic degeneracies. Whenever the microscopic
degeneracies are known the partition functions can be evaluated directly. This
is the case for N=4 heterotic CHL black holes, where we demonstrate that the
partition functions are consistent with the results obtained on the macroscopic
side for black holes that have a non-vanishing classical area. In this way we
confirm the presence of a measure in the duality invariant inverse Laplace
integrals. Most, but not all, of these results are obtained in the context of
semiclassical approximations. For black holes whose area vanishes classically,
there remain discrepancies at the semiclassical level and beyond, the nature of
which is not fully understood at present.
