This book integrates both classical and modern treatments of difference equations. It contains the most updated and comprehensive material on stability, Z-transform, discrete control theory, and asymptotic theory, continued fractions and orthogonal polynomials. Yet the presentation is simple enough for the book to be used by 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, epidemic models and host-parasitoid systems. Each section ends with an extensive and highly selected set of exercises.