  The core of this article is a general theorem with a large number of
specializations. Given a manifold $N$ and a finite number of one-parameter
groups of point transformations on $N$ with generators $Y, X_{(1)}, \cdots,
X_{(d)} $, we obtain, via functional integration over spaces of pointed paths
on $N$ (paths with one fixed point), a one-parameter group of functional
operators acting on tensor or spinor fields on $N$. The generator of this group
is a quadratic form in the Lie derivatives $\La_{X_{(\a)}}$ in the
$X_{(\a)}$-direction plus a term linear in $\La_Y$.
  The basic functional integral is over $L^{2,1}$ paths $x: {\bf T} \ra N$
(continuous paths with square integrable first derivative). Although the
integrator is invariant under time translation, the integral is powerful enough
to be used for systems which are not time translation invariant. We give seven
non trivial applications of the basic formula, and we compute its semiclassical
expansion.
  The methods of proof are rigorous and combine Albeverio H\oegh-Krohn
oscillatory integrals with Elworthy's parametrization of paths in a curved
space. Unlike other approaches we solve Schr\"odinger type equations directly,
rather than solving first diffusion equations and then using analytic
continuation.
