In this work we develop a general theory of oscillatory integrals on real Hilbert spaces and apply it to the mathematical foundation of the so called Feynman path integrals of non relativistic quantum mechanics, quantum statistical mechanics and quantum field theory. The translation invariant integrals we define provide a natural extension of the theory of finite dimensional oscillatory integrals, which has newly undergone an impressive development, and appear to be a suitable tool in infinite imensional analysis. For one example, on the basis of the present work we have extended the methods of stationary phase, Lagrange immersions and orresponding asymptotic expansions to the infinite dimensional case, covering in particular the expansions around the classical limit of quantum mechanics. A particular case of the oscillatory integrals studied in the present work are the Feynman path integrals used extensively in the physical literature, starting with the basic work on quantum dynamics by Dirac and Feynman, in the forties.