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),...,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 LX(α) in the X(α)-direction plus a term linear in LY. The basic functional integral is over L2,1 paths x:T→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. Seven nontrivial applications of the basic formula are given, and the semiclassical expansion is computed. The methods of proof are rigorous and combine Albeverio-Ho/egh-Krohn oscillatory integrals with Elworthy's parametrization of paths in a curved space. Unlike other approaches, Schrödinger type equations are solved directly, rather than solving first diffusion equations and then using analytic continuation.