  Functional integrals can be defined on topological groups in terms of
families of locally compact topological groups and their associated
Banach-valued Haar integrals. The definition forgoes the goal of constructing a
genuine measure on a space of functions, and instead provides for a topological
realization of localization in the infinite-dimensional domain. This yields
measurable subspaces that characterize meaningful functional integrals and a
scheme that possesses significant potential for representing non-commutative
Banach algebras suitable for mathematical physics applications. The framework
includes, within a broader structure, other successful approaches to define
functional integrals in restricted cases, and it suggests new and potentially
useful functional integrals that go beyond the standard Gaussian case. In
particular, functional integrals based on skew-Hermitian and K\"{a}hler
quadratic forms are defined and developed. Also defined are gamma-type and
Poisson-type functional integrals based on linear forms suggested by the gamma
probability distribution. These are expected to play an important role in
generating $C^\ast$-algebras of quantum systems. Several applications and
implications are presented.
