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