The problem of understanding path integrals associated to quantum gauge theories is a longstanding issue in mathematical physics, and now also in differential geometry. We show that quantum gauge theories in three and four dimensions are equivalent to purely fermionic theories, where, with appropriate cutoffs, the perturbation series is convergent. Classical techniques, developed in the 1980's, have been used in the past to understand the path measures in similar cases, and we hope that they are useful in this situation also. Meantime as a byproduct we obtain some natural conjectures about the behavior of correlations in four-dimensional Yang Mills theory, and explore connections with the invariants of three-manifolds. The technique of fermionization was developed in the 1970's in the context of two dimensional quantum field theories, where the fermions are quantum solitons or vertex operators. The fact that these ideas can be used in gauge theory may be indicative of the surprising ways in which integrals on infinite-dimensional spaces differ from their finite-dimensional analogs. In this talk I will survey the basic ideas of integration on function spaces and then discuss these new developments.