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A classification of discrete integrable systems on quad-graphs, i.e. onsurface cell decompositions with quadrilateral faces, is given. The notion ofintegrability laid in the basis of the classification is the three-dimensionalconsistency. This property yields, among other features, the existence of thediscrete zero curvature with a spectral parameter. For all integrable systemsof the obtained exhaustive list, the so called three-leg forms are found. Thisestablishes Lagrangian and symplectic structures for these systems, and theconnection to discrete systems of the Toda type on arbitrary graphs.Generalizations of these ideas to the three-dimensional integrable systems andto the quantum context are also discussed

Classification of Integrable Equations on Quad-Graphs. The Consistency Approach