This books aims to provide students and rsearchers with a basis for understanding the wide range of wave phenomena with which any mathematician may be confronted in applications. Compressible flow is the main focus of the book however the authors show how wave phenomena in electromagentism and solid mechanics can be treated using similar mathematical methods. This book originated from a course at Oxford University and previous book by H Ockendon and R Taylor entitled Inviscid Fluid Flows. This monograph has been retitled and revised throughly to reflect scientific interest. The book has exercises at end of each chapter and should appeal to senior undergraduate and graduate students interested in fluid mechanics.