Distinction is made between shallow water and deep water wave theories, depending on the value of the Ursell number. Potential flow theory is applied and the Stokes development is followed and first-order (linear), second-order, and third-order wave theories, in regular and irregular waves, are described. The concepts of phase and group velocities, mass transport, and energy flux are introduced. The application of stretching models to wave crest kinematics is described. At second-order distinction is made between bound (or locked) wave components (that accompany the first-order wave system) and free components (that travel independently). It is emphasized that, from third-order, such phenomena as mutual modifications of the phase velocities, or exchanges of energy, can take place between wave components. These interactions may lead to the occurrence of rogue waves, or to strong runups often observed at midships. The stream function wave model, which encompasses shallow and deep water cases, is presented. Finally the nonlinear Schr¨odinger equation that describes the time and space evolution of the wave envelope is applied to the prediction of the Benjaminis–Feir instability.