  The heat-kernel expansion and $\zeta$-regularization techniques for quantum
field theory and extended objects on curved space-times are reviewed. In
particular, ultrastatic space-times with spatial section consisting in manifold
with constant curvature are discussed in detail. Several mathematical results,
relevant to physical applications are presented, including exact solutions of
the heat-kernel equation, a simple exposition of hyperbolic geometry and an
elementary derivation of the Selberg trace formula. With regards to the
physical applications, the vacuum energy for scalar fields, the one-loop
renormalization of a self-interacting scalar field theory on a hyperbolic
space-time, with a discussion on the topological symmetry breaking, the finite
temperature effects and the Bose-Einstein condensation, are considered. Some
attempts to generalize the results to extended objects are also presented,
including some remarks on path integral quantization, asymptotic properties of
extended objects and a novel representation for the one-loop (super)string free
energy.
