There is a certain body of knowledge and methods that finds application in most branches of fluid mechanics. This book aims to supply a proper theoretical understanding that will permit sensible simplifications to be made in the formulation of problems, and enable the reader to develop analytical models of practical significance. Such analyses can be used to guide more detailed experimental and numerical investigations. As in most technical subjects, such understanding is acquired by detailed study of highly simplified 'model problems'. The first part (Chapters 1 - 4) is concerned entirely with the incompressible flow of a homogeneous fluid. It was written for the Boston University introductory graduate level course 'Advanced Fluid Mechanics'. The remaining Chapters 5 and 6 deal with dispersive waves and acoustics, and are unashamedly inspired by James Lighthill's masterpiece, Waves in Fluids.