  We review localization techniques for functional integrals which have
recently been used to perform calculations in and gain insight into the
structure of certain topological field theories and low-dimensional gauge
theories. These are the functional integral counterparts of the Mathai-Quillen
formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula
respectively. In each case, we first introduce the necessary mathematical
background (Euler classes of vector bundles, equivariant cohomology, topology
of Lie groups), and describe the finite dimensional integration formulae. We
then discuss some applications to path integrals and give an overview of the
relevant literature. The applications we deal with include supersymmetric
quantum mechanics, cohomological field theories, phase space path integrals,
and two-dimensional Yang-Mills theory.
