Transition to Chaos in Classical and Quantum Mechanics : Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme Italy July 6–13 1991 The Session was intended to give a broad survey of the mathematical problems arising in the chaotic transition of deterministic dynamical systems both in classical and quantum mechanics. The lectures of Mather and Forni thoroughly cover the area preserving twist maps and include an up to date version of the Aubry Mather theory. The lectures of Bellissard describe the quantum aspects: classical limit localization spectral properties of the relevant Schrödinger operators and thus represent an exhaus tive introduction to the mathematics of quantum chaos . The lectures of Degli Esposti et al. reviews equidistribu tion of unstable periodic orbits and classical limit of quantized toral symplectomorphisms