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We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity.doi:10.1098/rsta.2017.0093 pmid:29229793 fatcat:lxsfixdqpvbhrklfclyr7uhvp4