Three-dimensional travelling gravity-capillary water waves

The travelling water-wave problem is one of the classical problems in applied mathematics, and there is a huge and growing literature in this area. However virtually all the published results on three-dimensional travelling water waves concentrate on numerical studies or approximations by simper model equations, and there has been far less rigorous mathematical study of the complete hydrodynamic problem. Functional-analytic techniques for partial differential equations are required to make progress with the exact equations, and in this paper we survey the currently available mathematical results.