  We study natural cluster structures in the rings of regular functions on
simple complex Lie groups and Poisson-Lie structures compatible with these
cluster structures. According to our main conjecture, each class in the
Belavin-Drinfeld classification of Poisson-Lie structures on $\G$ corresponds
to a cluster structure in $\O(\G)$. We prove a reduction theorem explaining how
different parts of the conjecture are related to each other. The conjecture is
established for $SL_n$, $n<5$, and for any $\G$ in the case of the standard
Poisson-Lie structure.
