This volume contains a series of lectures 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, and spectral properties of the relevant Schrodinger operators, thus representing an exhaustive introduction to the mathematics of "quantum chaos". Other lectures review equidistribution of unstable periodic orbits and the classical limit of quantized total symplectomorphisms.