Discrete systems and integrability This first introductory text to Discrete Integrable Systems introduces key notions of integrability from the vantage point of discrete systems also making connections with the continuous theory where relevant. While treating the material at an elementary level the book also highlights many recent developments. Topics include: Darboux and Bäcklund transformations difference equations and special functions multidimensional consistency of integrable lattice equations associated linear problems (Lax pairs) connections with Padé approximants and convergence algorithms Hirotas bilinear formalism for lattices intriguing properties of discrete Painlevé equations and the novel theory of Lagrangian multiforms. The book builds the material in an organic way emphasizing interconnections between the various approaches while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field the numerous exercises and the thorough list of references will benefit upper level undergraduate and beginning graduate students as well as researchers from other disciplines