Cluster integrable systems, q-Painlevé equations and their quantization

We discuss the relation between the cluster integrable systems and q-difference Painleve equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painleve dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painleve systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painleve system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.