In this talk I would like to report on the progress in the conjecture, that each class in the Belavin-Drinfeld classification of Poisson-Lie structures on a simple complex Lie group $G$ corresponds to a cluster structure in the ring of regular funcions $\mathcal O(G)$ for $G=SL_n$.

This is a joint work with M. Gekhtman and A. Vainshtein.