Cluster structures were discovered by S. Fomin and A. Zelevinsky about twenty years
ago and quickly found applications in various fields of mathematics and mathematical physics.
In the latter, several advances were made in a study of classical and quantum integrable
systems arising in the context of cluster structures. These systems "live" on Poisson-Lie
groups and their Poisson homogeneous spaces, hence it is important to understand an
interplay between cluster and Poisson structures on such objects.
In this talk I will explain a construction of a family of (generalized) cluster structures in the
algebra of regular functions on SL_n related to the Belavin-Drinfeld classification
of Poisson-Lie structures on SL_n.
Based on a joint work with M.~Gekhtman (Notre Dame) and M.~Shapiro (MSU).