Transport phenomena are broad, diverse subjects with many specific areas, including compressible single-fluid, two-fluid, and two-phase fluid dynamics with phase change, chemical reactions, and heat transfer. This chapter discusses the mathematical models of compressible single-fluid dynamics and develop insights into their solutions and the physics, which will, in turn, guide the design of high-fidelity numerical algorithms for such models, as will be discussed in later chapters. Two-fluid and two-phaseTwo-phase flows share similar characteristics with the single-fluid flows but have their unique complexities, which will be discussed starting in Chap.  6 and continuing through Chap.  9 .