Abstract : The dynamics of waves in deep water and in shallow water are quite different. In shallow water, we, have shown experimentally the existence of a family of finite-amplitude waves that propagate practically without change of form in shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional, and periodic. The waves are easy to generate experimentally, and they seem to be stable to perturbations. They are described with reasonable accuracy by an 8-parameter family of exact solutions of the Kadomtsev-Petviahsvili equation. The situation is quite different in deep water, where much of our knowledge is based on numerical simulations. An approximate model of one-dimensional, nearly monochromatic waves. in deep water is the nonlinear Schrodinger equation. We find that for certain ranges of parameters in initial data, numerical solutions of the equation are so unstable that long-time simulations of the equation are not reproducible, and are quite unreliable.