Oscillatory Integrals and Phenomena Beyond all Algebriac Orders : With Applications to Homoclinic Orbits in Reversible Systems During the last two decades in several branches of science (water waves crystal growth travelling waves in one dimensional lattices splitting of separatrices ..) different problems appeared in which the key point is the computation of exponentially small terms. This self contained monograph gives new and rigorous mathematical tools which enable a systematic study of such problems. Starting with elementary illuminating examples the book contains (i) new asymptotical tools for obtaining exponentially small equivalents of oscillatory integrals involving solutions of nonlinear differential equations; (ii) implementation of these tools for solving old open problems of bifurcation theory such as existence of homoclinic connections near resonances in reversible systems