The Springer edition of this book is an unchanged reprint of Courant and Friedrich’s classical treatise which was first published in 1948. The basic research for it took place during World War II, but there are many aspects which still make the book interesting as a text and as a reference. It treats basic aspects of the dynamics of compressible fluids in mathematical form, and attempts to present a systematic theory of nonlinear wave propagation, particularly in relation to gas dynamics. Written in the form of an advanced textbook, it accounts for classical as well as some fairly recent developments. The authors intend their audience to be engineers, physicists and mathematicians alike.