i will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. i will begin by giving a preciese account of the formulation of the surface water-wave problem and discusion of its distinct features. they include the dispersion relation, its severe nonlinearity, traveling waves and the hamiltonian structure. i will describe the recent work of hans christianson, gigliola staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. if time permits, i will explore some oen questions regarding long-time behavior and stability.