One aim of this report is to show and emphasize that in the computational approaches to most of today's pressing and challenging scientific and technological problems, the mathematical aspects cannot and should not be considered in isolation. Following an introductory chapter, chapter 2 discusses a number of typical problems leading to computational modeling from the viewpoint of the scientific or engineering discipline in which they arise. Areas considered include: hydrodynamic systems; chemical systems and combustion; plasma physics; particle physics; condensed-matter physics; geophysical applications; meteorology; astrophysics; structural mechanics; nondestructive testing and tomographic reconstruction; mathematical models in the biological sciences; and electronic components. Chapter 3 examines several of the problems from the viewpoint of the computational and mathematical difficulties that arise in connection with them. These difficulties include larger numbers of degrees of freedom, different time or length scales, or singularities of various types. Chapter 4 focuses on the numerical algorithms involved in the computations, such as various discretization methods, continuation approaches, and splitting techniques. Increased support for computational modeling and applied mathematics, computing facilities, and for education and manpower development in these areas is recommended. (JN)