  In this thesis, we develop path integral localization methods that are
familiar from topological field theory: the integral over the infinite
dimensional integration domain depends only on local data around some finite
dimensional subdomain.
  We introduce a new localization principle that unifies BRST localization, the
non-Abelian localization principle and the conformal generalization of the
Duistermaat-Heckman integration formula.
  In addition, it is studied if one can possibly derive a generalized Selberg's
trace formula on locally homogeneous manifolds using localization techniques.
However, a definite answer is obtained only in the Lie group case (we complete
the work of R. Picken) in which it is an application of the Duistermaat-Heckman
integration formula. Also a new derivation of DeWitt's term is reported.
  Furthermore, connections between evolution operators of integrable models and
localization methods are studied. A derivative expansion localization is
presented and it is conjectured to apply also to integrable models, for example
the Toda lattice.
  Moreover, a pedagogical introduction to the localization techniques is given,
as well as a list of selected references that might be useful for a beginning
graduate student in mathematical physics or for a mathematician who would like
to study the physical point of view to topological field theory and string
theory.
