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where progress at any moment can be regarded as a standard to measure the real success of mathematical science’. While the study of water waves draws freely from other disciplines, the development of the subject has proved that one cannot expect themathematical tools to have been laid out and only be usedmechanically. Even at simple levels, problems in hydrodynamics require subtle use of mathematical concepts in order to obtain effective solutions that reflect to some extent the physical reality. Indeed, ‘.hydrodynamical theories stimulateddevelopments in thedomains of pure mathematics, such as complex analysis, topology. ’ (cf. Arnold & Khesin 1998). Despite the great progress achieved in hydrodynamics in the last two centuries, the subject still offers plenty of fertile ground for further exploration, providing in many fundamental aspects compelling challenges for researchers. Starting with the work of Stokes and his contemporaries in the nineteenth century (Craik 2004), there have been significant advances in our understanding of water waves. Until the second half of the twentieth century, the study of water wa On *athematics, but also a considerable part of themathematical theory ofwavemotio r the past 200 years has been pioneered on the basis of studies of water waves. TIntroduction: some recent developments of nonlinear water wave theor