  An Euclidean approach for investigating quantum aspects of a scalar field
living on a class of D-dimensional static black hole space-times, including the
extremal ones, is reviewed. The method makes use of a near horizon
approximation of the metric and $\zeta$-function formalism for evaluating the
partition function and the expectation value of the field fluctuations
$<\phi^2(x)>$. After a review of the non-extreme black hole case, the extreme
one is considered in some details. In this case, there is no conical
singularity, but the finite imaginary time compactification introduces a cusp
singularity. It is found that the $\zeta$-function regularized partition
function can be defined, and the quantum fluctuations are finite on the
horizon, as soon as the cusp singularity is absent, and the corresponding
temperature is T=0.
