  We introduce renormalized integrals which generalize conventional measure
theoretic integrals. One approximates the integration domain by measure spaces
and defines the integral as the limit of integrals over the approximating
spaces. This concept is implicitly present in many mathematical contexts such
as Cauchy's principal value, the determinant of operators on a Hilbert space
and the Fourier transform of an $L^p$-function.
  We use renormalized integrals to define a path integral on manifolds by
approximation via geodesic polygons. The main part of the paper is dedicated to
the proof of a path integral formula for the heat kernel of any self-adjoint
generalized Laplace operator acting on sections of a vector bundle over a
compact Riemannian manifold.
