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This Research Note sets forth a comprehensive description of various aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics , shallow water, and solid dynamics equations. This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes it valuable not only to those interested the development of numerical methods, but also to physicists and engineers.