This book combines both analytic and geometric (topological) approaches to studying difference equations. It integrates both classical and modern treatment of the subject. The book has the most updated and comprehensive material in stability, z-transform, discrete control theory, and asymptotic theory. Yet the presentation is simple enough so that the book can be used for advanced undergraduate and beginning graduate students in mathematics, engineering, science, and economics. Moreover, scientists and engineers who are interested in discrete mathematical models will find it useful as a reference. The book contains a large set of applications in a variety of disciplines including neural networks, feedback control, Markov chains, trade models, heat transfer, propagation of plants, etc. Each section of the book ends with an extensive and highly selected set of exercises.