This book is an introduction to a fascinating topic at the interface of functional analysis, algebra and numerical analysis, written for a broad audience of students, researchers and practitioners. It is concerned with the study of infinite matrices and their approximation by matrices of finite size. Our framework includes the simplest, important case where the matrix entries are numbers, but also the more general case where the entries are bounded linear operators. This ensures that examples of the class of operators studied - band-dominated operators on Lebesgue function spaces and sequence spaces - are ubiquitous in mathematics and physics. The main items and concepts studied are band-dominated operators, invertibility at infinity, Fredholmness, the method of limit operators, and the stability and convergence of finite matrix approximations. Concrete examples are used to illustrate the results throughout, including discrete Schrödinger operators and integral and boundary integral operators arising in mathematical physics and engineering. The main audience for this book are people concerned with large finite matrices and their infinite counterparts, for example in numerical linear algebra and mathematical physics. More generally, the book will be of interest to those working in operator theory and applications, for example studying integral operators or the application of operator algebra methods. While some basic knowledge of functional analysis would be helpful, the presentation contains relevant preliminary material and is largely self-contained.