  We review equivariant localization techniques for the evaluation of Feynman
path integrals. We develop systematic geometric methods for studying the
semi-classical properties of phase space path integrals for dynamical systems,
emphasizing the relations with integrable and topological quantum field
theories. Beginning with a detailed review of the relevant mathematical
background -- equivariant cohomology and the Duistermaat-Heckman theorem, we
demonstrate how the localization ideas are related to classical integrability
and how they can be formally extended to derive explicit localization formulas
for path integrals in special instances using BRST quantization techniques.
Various loop space localizations are presented and related to notions in
quantum integrability and topological field theory. We emphasize the common
symmetries that such localizable models always possess and use these symmetries
to discuss the range of applicability of the localization formulas. A number of
physical and mathematical applications are presented in connection with
elementary quantum mechanics, Morse theory, index theorems, character formulas
for semi-simple Lie groups, quantization of spin systems, unitary integrations
in matrix models, modular invariants of Riemann surfaces, supersymmetric
quantum field theories, two-dimensional Yang-Mills theory, conformal field
theory, cohomological field theories and the loop expansion in quantum field
theory. Some modern techniques of path integral quantization, such as coherent
state methods, are also discussed. The relations between equivariant
localization and other ideas in topological field theory, such as the
Batalin-Fradkin-Vilkovisky and Mathai-Quillen formalisms, are presented.
