Mathematics and the sciences have grown up together, repeatedly interacting. Discoveries in science open up new advances in statistics, computer science, operations research, and pure and applied mathematics. These, in turn, enable new practical technologies and advance entirely new frontiers of science. Frequently, however, cooperation and collaboration between mathematical scientists and scientists have come as a result of chance encounters. Often a scientific or technological problem exists for years before a mathematical scientist discovers it and recognizes it as interesting and mathematically tractable. Similarly, scientists have frequently expended energy “inventing” mathematical solutions that existed for decades but were not familiar to them from their training. To encourage new linkages between the mathematical sciences and other fields and to sustain old linkages, the National Research Council (NRC) constituted a committee representing a broad cross-section of scientists from academia, federal government laboratories, and industry. Its task was to “examine the mechanisms for strengthening interdisciplinary research between the sciences and mathematical sciences, with the principal efforts being to suggest what are likely to be the most effective mechanisms for collaboration. . . . ” The committee believes that the benefits to be derived from cross-disciplinary activities linking the mathematical sciences with science and engineering are indeed huge and far reaching.