This book reviews localization for the equivariant evaluation of techniques It Feynmanpathintegrals. methods for developssystematicgeometric study the semi classical of ing for properties phase spacepath integrals dynamical the relationswith and systems, emphasizing integrable topologicalquantum field theories. with a detailed review ofthe relevant Beginning mathemat ical and the Duistermaat Heckman background equivariant cohomology it demonstrates theorem, how the localization ideas are related to classical andhow be can extended to derive local integrability they formally explicit izationformulas for in instances path BRST integrals special using quantiza tion Various localizations techniques. loop are andrelatedto space presented notions in and field quantum Thebookem integrability topological theory. thecommon thatsuchlocalizablemodels phasizes symmetries always possess and uses these to discussthe of ofthe local symmetries range applicability ization formulas.Anumber of and mathematical are physical applications inconnectionwith presented Morse elementaryquantummechanics, theory, index character formulas for Lie theorems, semi simple groups, quantization of inmatrix modular spinsystems, unitaryintegrations models, invariantsof Riemarm field two dimensional surfaces, supersymmetricquantum theories, conformal field field theories and Yang Mills theory, theory, cohomological the in field Somemodern loopexpansion of quantum theory. techniques path such coherent state also integral quantization, as are discussed. methods, The relationsbetween localizationandother ideas in equivariant topological field such asthe and theory, for Batalin Fradkin Vilkovisky Mathai Quillen are and used to discuss the malisms, presented between general relationship field theoriesandmore conventional models. topological physical Copenhagen, 1999 Richard J. Szabo September Iwould like to thank I. G. F. Acknowledgements: Kogan, Landi, Lizzi and G.