Many federal funding requests for more advanced computer resources assume implicitly that greater computing power creates opportunities for advancement in science and engineering. This has often been a good assumption. Given stringent pressures on the federal budget, the White House Office of Management and Budget (OMB) and Office of Science and Technology Policy (OSTP) are seeking an improved approach to the formulation and review of requests from the agencies for new computing funds. This book examines, for four illustrative fields of science and engineering, how one can start with an understanding of their major challenges and discern how progress against those challenges depends on high-end capability computing (HECC). The four fields covered are: * atmospheric science * astrophysics * chemical separations * evolutionary biology This book finds that all four of these fields are critically dependent on HECC, but in different ways. The book characterizes the components that combine to enable new advances in computational science and engineering and identifies aspects that apply to multiple fields.