  We consider general integrable systems on graphs as discrete flat connections
with the values in loop groups. We argue that a certain class of graphs is of a
special importance in this respect, namely quad-graphs, the cellular
decompositions of oriented surfaces with all two-cells being quadrilateral. We
establish a relation between integrable systems on quad-graphs and discrete
systems of the Toda type on graphs. We propose a simple and general procedure
for deriving discrete zero curvature representations for integrable systems on
quad-graphs, based on the principle of the three-dimensional consistency. Thus,
finding a zero curvature representation is put on an algorithmic basis and does
not rely on the guesswork anymore. Several examples of integrable systems on
quad-graphs are considered in detail, their geometric interpretation is given
in terms of circle patterns.
