Recent interactions between the fields of geometry, classical and quantum dynamical systems, and the visualization of geometric objects have led to the observation that most concepts of surface theory and of the theory of integrable systems have natural discrete analogues. For these analogues the corresponding difference equations are often integrable, which has in turn led to important results in such areas as condensed matter physics and quantum field theory. This book combines the efforts of a distinguished team of authors from various fields in mathematics and physics to provide an accessible overview of the subject. The book begins with the mathematical concepts of discrete geometry and discrete integrable systems, which are interesting on their own, and then proceeds to develop the many connections with classical and quantum dynamics.