This collection of papers is a novel publication for SIAM. It is, first and foremost, a forum for leading industrial and government scientists and engineers to describe their work and what they hope to accomplish over the next decade. The goal is to focus the mathematical community on research that will both advance mathematics and provide immediate benefits to industry. So this is not a book on mathematics in the strictest sense. It is not organized around a single or even multiple mathematical disciplines. The papers contain few, if any, statements and proofs of theorems. Rather, they focus on describing the needs of industry and government and on highlighting mathematics that may play a role in providing solutions. The papers were elicited primarily from Mathematics for Industry: Challenges and Frontiers, a conference sponsored by SIAM in October, 2003. This conference was different from other SIAM conferences in that it was a forum for scientific experts from industry and government to meet with members of the mathematical community to discuss the visions they had for their industries and the obstacles they saw to reaching those goals. A distinctive feature of this work is its organization. The three sections: Industrial Problems; Mathematical Responses to Industrial Problems; and The Process of Mathematical Modeling, Practice, and Education offer a process view, always initiated by a practical problem, leading to the need for responsive mathematics both at the research and educational levels. All topics are tightly integrated with contemporary computing technology. Mathematical topics include splines, modeling, computational fluid dynamics, statistical model verification, and mathematics for design and simulation. Academic mathematicians who wish to broaden their view of industrial mathematics will find this book useful. Industrial mathematicians will find support for the increasing need to collaborate with academic mathematicians and scientists. Students who are, or are thinking of, pursuing a career in industrial or academic applied mathematics will find this book a rich source of material on the problems facing industry and will develop an understanding of the relationship between mathematics and industrial problems. Teachers of applied mathematics can use this book to augment standard teaching materials.