  We consider the extremal limit of a black hole geometry of the
Reissner-Nordstrom type and compute the quantum corrections to its entropy.
Universally, the limiting geometry is the direct product of two 2-dimensional
spaces and is characterized by just a few parameters. We argue that the quantum
corrections to the entropy of such extremal black holes due to a massless
scalar field have a universal behavior. We obtain explicitly the form of the
quantum entropy in this extremal limit as function of the parameters of the
limiting geometry. We generalize these results to black holes with toroidal or
higher genus horizon topologies. In general, the extreme quantum entropy is
completely determined by the spectral geometry of the horizon and in the
ultra-extreme case it is just a determinant of the 2-dimensional Laplacian. As
a byproduct of our considerations we obtain expressions for the quantum entropy
of black holes which are not of the Reissner-Nordstrom type: the extreme
dilaton and extreme Kerr-Newman black holes. In both cases the classical
Bekenstein-Hawking entropy is modified by logarithmic corrections.
