Instabilities of Stokes waves

In 1847 Stokes laid out the first construction of exact solutions to the nonlinear water wave problem. They are unidirectional traveling waves with periodic profiles of small amplitude. Experiments and informal analysis initiated by Benjamin and Feir in 1967 suggested that Stokes waves are unstable. Their work has led to a vast literature on numerical and informal analysis for various types of instabilities of Stokes waves, and the last six years have seen successful rigorous proofs of some of these instabilities. This talk is devoted to a survey of these rigorous results.