This carefully written and well-thought-out book presents a comprehensive view on mathematical theory concerning multi-dimensional hyperbolic partial differential equations. The main body of the book consists of four parts. The first part deals with the theory of linear Cauchy problems that involve both constant and variable coefficients, the latter illustrating the power of pseudo-differential and para-differential calculus (presented separately in the appendix). The second part is devoted to linear initial boundary value problems, proceeding from simpler to more complicated systems (symmetric, constant coefficients, variable coefficients). The third part is devoted to the theory of nonlinear problems, focusing on the notion of a smooth solution and a piecewise smooth solution suitable for analysis of shock waves (as is carefully proved). The last part investigates problems in gas dynamics and it includes a discussion of appropriate boundary conditions and shock-wave analysis. The text is completed with an extensive bibliography including classical and recent papers both in partial differential equation analysis and applications (mainly in gas dynamics).