This text is based on lecture courses given by the author, over about 40 years, at Newcastle University, to final-year applied mathematics students. It has been written to provide a typical course that introduces the majority of the relevant ideas, concepts and techniques, rather than a wide-ranging and more general text. Thus the topics, with their detailed discussion linked to the many carefully worked examples, do not cover as broad a spectrum as might be found in other, more wideranging texts on fluid mechanics; this is a quite deliberate choice here. Thus the development follows that of a conventional introductory module on fluids, comprising a.