The first chapter of this book provides a brief treatment of the basics of the subject. The other chapters deal with the various decompositions of non-negative matrices, Birkhoff type theorems, the study of the powers of non-negative matrices, applications of matrix methods to other combinatorial problems, and applications of combinatorial methods to matrix problems and linear algebra problems. The coverage of prerequisites has been kept to a minimum. Nevertheless, the book is basically self-contained (an Appendix provides the necessary background in linear algebra, graph theory and combinatorics). There are many exercises, all of which are accompanied by sketched solutions. Audience: The book is suitable for a graduate course as well as being an excellent reference and a valuable resource for mathematicians working in the area of combinatorial matrix theory.