  Equivariant localization theory is a powerful tool that has been extensively
used in the past thirty years to elegantly obtain exact integration formulas,
in both mathematics and physics. These integration formulas are proved within
the mathematical formalism of equivariant cohomology, a variant of standard
cohomology theory that incorporates the presence of a symmetry group acting on
the space at hand. A suitable infinite-dimensional generalization of this
formalism is applicable to a certain class of Quantum Field Theories (QFT)
endowed with supersymmetry. In this thesis we review the formalism of
equivariant localization and some of its applications in Quantum Mechanics (QM)
and QFT. We start from the mathematical description of equivariant cohomology
and related localization theorems of finite-dimensional integrals in the case
of an Abelian group action, and then we discuss their formal application to
infinite-dimensional path integrals in QFT. We summarize some examples from the
literature of computations of partition functions and expectation values of
supersymmetric operators in various dimensions. For 1-dimensional QFT, that is
QM, we review the application of the localization principle to the derivation
of the Atiyah-Singer index theorem applied to the Dirac operator on a twisted
spinor bundle. In 3 and 4 dimensions, we examine the computation of expectation
values of certain Wilson loops in supersymmetric gauge theories and their
relation to 0-dimensional theories described by "matrix models". Finally, we
review the formalism of non-Abelian localization applied to 2-dimensional
Yang-Mills theory and its application in the mapping between the standard
"physical" theory and a related "cohomological" formulation.
