Cazenave (Pierre and Marie Curie University) reviews the fundamental properties of the linear Schrodinger equation and some results of local existence for the nonlinear Schrodinder equation in a general domain, then examines the local Cauchy problem, the regularity of solutions, the smoothing effect, global existence, finite-time blowup of solutions, the asymptotic behavior of solutions in the repulsive case, and the stability of standing waves in the attractive case.