The variety of combinatorial properties of nonnegative matrices is widely discussed in the mathematical literature, and there are many papers on this topic. However, there are few monographs devoted to these properties of nonnegative matrices. This book fills that gap and presents a summary of the existing material. It provides a good entry point into the subject and includes exercises to aid students. The authors focus on the relation of matrices with nonnegative elements to various mathematical structures studied in combinatorics. In addition to applications in graph theory, Markov chains, tournaments, and abstract automata, the authors consider relations between nonnegative matrices and structures such as coverings and minimal coverings of sets by families of subsets. They also give considerable attention to the study of various properties of matrices and to the classes formed by matrices with a given structure. The authors discuss enumerative problems using both combinatorial and probabilistic methods. It also considers extremal problems related to matrices and problems where nonnegative matrices provide suitable investigative tools. This book was developed for the most part as a theoretical research text, keeping in mind applications of nonnegative matrices. Among the applications, the most significant included are in the theory of Markov chains, in linear programming for constructing and analyzing economic models, and in information theory for designing reliable information devices. The book is suitable for specialists in these areas of engineering and the applied sciences. The book contains some classical theorems and a significant number of results not previously published in monograph form, including results obtained by the authors in the last few years. It is appropriate for graduate students and researchers interested in combinatorics and its applications.