High resolution upwind and centred methods are a generation of computational techniques applicable to a wide range of engineering and scientific disciplines, Computational Fluid Dynamics (CFD) being the most prominent. This textbook provides a presentation of this class of techniques. It is designed to provide readers with an understanding of the basic concepts and some of the underlying theory, the ability to use research papers on the subject and the information required for the practical implementation of the methods. Applications include: compressible, steady, unsteady, reactive, viscous, non-viscous and free surface flows.