This book provides an integrated treatment of the theory of nonnegative matrices and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimization and mathematical economics. The authors have chosen the wide variety of applications, which include price fixing, scheduling, and the fair division problem, both for their elegant mathematical content and for their accessibility to students with minimal preparation. They present many new results in matrix theory for the first time in book form, while they present more standard topics in a novel fashion. The treatment is rigorous and almost all results are proved completely. These new results and applications will be of great interest to researchers in linear programming, statistics, and operations research. The minimal prerequisites also make the book accessible to first year graduate students.