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This monograph presents the existence and properties of both weak and strong solutions to the problems of the flow of a compressible fluid in a domain whose motion is prescribed. Chapters build upon the research of Lions and Feireisl with regards to weak solutions to the compressible version of the Navier-Stokes system, and extend it to problems on moving domains. The authors also show the existence of strong solutions to the compressible Navier-Stokes system for either a small time interval or small data. The opening chapters introduce the notation, tools, and problems covered in the rest of the book, emphasizing pedagogy and accessibility throughout. Mathematical Theory of Compressible Fluids on Moving Domains will be suitable for graduate students and researchers interested in mathematical fluid mechanics.