The phenomenon of disintegration of wave trains, commonly known as the Benjamin-Feir instability, on deep as well as shallow water is reviewed covering both the original perturbation theory of Benjamin and Feir [4] as well as the alternative approach through the nonlinear Schrödinger equation, such as that given by Stuart and DiPrima [17]. Also presented is a short description of higher-order instability of wave trains whose astonishing result is due to the discovery made by Longuet-Higgins [9, 10]. The extension of the perturbation theory to instability of three-dimensional oblique waves [1] is also briefly explained, and a compelling suggestion is given for a possible application of this theory to fully three-dimensional waves on deep water.

Benjamin-Feir instability, water waves, three-dimensional waves.