The study of fluid mechanics is fundamental to modern applied mathematics, with applications to oceans, the atmosphere, flow in pipes, aircraft, blood flow and very much more. This text provides an introduction to the mathematical approach to this subject and to many of its main ideas, based on material typically found in most university courses. So, firstly, the methods and fundamental results, for a general fluid, are presented, and, secondly, the lift generation of aerofoils is analysed by using complex potentials. Numerous worked examples are provided throughout, as are many set exercises.