The theory of water waves has been a source of intriguing – and often difficult – mathematical problems for at least 150 years. Virtually every classical mathematical technique appears somewhere within its confines; in addition, linear problems provide a useful exemplar for simple descriptions of wave propagation, with nonlinearity adding an important level of complexity. It is, perhaps, the most readily accessible branch of applied mathematics, which is the first step beyond classical particle mechanics. It embodies the equations of fluid mechanics, the concepts of wave propagation, and the critically important rôle of boundary conditions. Furthermore, the results of a calculation provide a description that can be tested whenever an expanse of water is to hand: a river or pond, the ocean, or simply the household bath or sink. Indeed, the driving force for many workers who study water waves is to obtain information that will help to tame this most beautiful, and sometimes destructive, aspect of nature. (Perhaps ‘to tame’ is far too bold an ambition: at least to try to make best use of our knowledge in the design of man-made structures.) Here, though, we shall – without apology – restrict our discussion to the many and varied aspects of water-wave theory that are essentially mathematical. Such studies provide an excellent vehicle for the introduction of the modern approach to applied mathematics: complete governing equations; nondimensionalisation and scaling; rational approximation; solution; interpretation. This will be the type of systematic approach that is adopted throughout this text.