One of the major achievements in computational fluid dynamics has been the development of numerical methods for simulating compressible flows. These methods combine higher-order accuracy in smooth regions, with sharp, oscillation-free representation of embedded shocks, and are now known as "high-resolution schemes". This volume collects in one place many of the most significant papers in the development of high-resolution schemes as occurred at ICASE, together with introductions from the editors, written from the modern vantage point. Chapter I covers the early development of approximate Riemann solvers in the context of monotonic schemes. Chapter II continues with papers dealing with second-order total-variation-diminishing (TVD) schemes. Chapter III presents papers which show the historical development of total-variation-bounded (TVB) schemes at ICASE. Finally, Chapter IV collects papers on inherently multidimensional schemes, which preserve the attractive features of their one-dimensional predecessors.