This concise volume by a prominent mathematician offers an important survey of mathematical aspects of the theory of compressible fluids. The treatment is geared toward advanced undergraduates and graduate students in physics, applied mathematics, and engineering. Focusing on two-dimensional steady potential flows, the text eschews detailed proofs in favor of clear indications of the main ideas and descriptions of new mathematical concepts and methods that arose in connection with these chapters in fluid dynamics. Starting with a general discussion of the differential equations of a compressible gas flow, the book advances to the mathematical background of subsonic flow theory. Subsequent chapters explore the behavior of a flow at infinity and methods for the determination of flows around profiles, flows in channels and with a free boundary, the mathematical background of transonic gas dynamics, and some problems in transonic flow. An extensive bibliography of 400 papers concludes the text.