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 threedimensional 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.

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on Z 2 . The fields are associated with the vertices and an equation of the form Q(x 1 , x 2 , x 3 , x 4 ) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices Z N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.

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Birational Yang-Baxter maps ('set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map (x, y) → (u, v) is called quadrirational, if its graph is also a graph of a birational map (x, v) → (u, y). We obtain a classification of quadrirational maps on CP 1 × CP 1 , and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.

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