The critical role of computers in scientific advancement is described in this panel report. With the growing range and complexity of problems that must be solved and with demands of new generations of computers and computer architecture, the importance of computational mathematics is increasing. Multidisciplinary teams are needed; these are found in most advanced and industrial laboratories, but rarely in universities. The existing educational opportunities are not producing the required personnel to meet substantial shortages. Therefore, the panel strongly recommends increased federal support for: (1) research in computational mathematics, methods, algorithms, and software for scientific computing; (2) the development of interdisciplinary research teams; (3) the establishment and continued operation of a suitable research infrastructure for the teams; (4) graduate and post-doctoral students directly involved in the research of some interdisciplinary team; and (5) young researchers and cross-disciplinary visitors. In the second section, research opportunities in a number of mathematical areas are described. New modes of research are discussed next, followed by comments on educational needs and a final section on funding considerations. Appendices contain a list of related reports, information on laboratory facilities for scientific computing, and letters and position papers. (MNS)