Chapter 2 Numerical Solution of Quasilinear Hyperbolic Systems In this chapter we describe basic approaches to constructing shock-capturing and shockfitting methods for solving multidimensional quasilinear hyperbolic systems of general form. Among numerical methods we mainly select those that are based on the exact or approximate solution of the corresponding one-dimensional Riemann problem of disintegration of an arbitrary discontinuity or can be interpreted as based on this solution. Such methods are called Godunov methods. They proved to be extremely fruitful in numerous applications. This is due to the fact that the Godunov-type methods are based on the fundamental properties of hyperbolic systems.