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 while keeping the book 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.