This book systematically studies upwind methods for initial value problems for scalar conservation laws in one- and multidimensions. The mathematical theory of convergence theory and of a priori error estimates is presented in detail for structure (finite difference methods) as well as for unstructured grids (finite volume methods). Higher order schemes are also included. In the second part of the book the algorithms for scalar equations are generalized into systems of conversation laws in one- and multidimensions. The most powerful schemes for the discretization of systems are described and numerical examples are presented. In particular, local grid refinement has been taken into account. The initial boundary value problem is also considered for linear systems and nonlinear scalar conservation laws.