The focus of this book is on non-negative matrices, exploring their diverse combinatorial properties that are widely discussed in mathematical literature and have inspired a significant number of articles. However, there is relatively little monographic literature dedicated to the combinatorial properties of non-negative matrices. The authors aim to shift the focus away from traditional algebraic and, in particular, spectral properties of these matrices, and instead highlight their connections to various mathematical structures that are central to the study of combinatorial mathematics. In addition to the conventional applications of non-negative matrices in graph theory, Markov chains, tournaments, and abstract automata, the book establishes links to non-negative matrices associated with objects such as coverings and minimal coverings of finite sets by their subsets. Alongside the exploration of combinatorial concepts interpreted through non-negative matrices, considerable attention is given to investigating the various properties of the matrices themselves, as well as classes that encompass matrices with specific structures. A significant portion of the study is devoted to examining the asymptotic properties of non-negative matrices as certain parameters that characterize the matrix grow without bound.