We will discuss geometric properties of double Bruhat cells and their totally positive parts in complex semisimple groups. The ideas and results to be discussed were influenced by the work of G.Lusztig and obtained jointly with A.Berenstein, S.Fomin, B. and M.Shapiro, and A.Vainshtein. These ideas provided the groundwork for the theory of cluster algebras.

Cluster algebras, introduced jointly with S.Fomin, are a class of axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of the same finite cardinality. The original motivation for this theory lied in the desire to create an algebraic framework for total positivity and canonical bases in semisimple algebraic groups. We will discuss the basics of the theory of cluster algebras and their quantum analogues constructed jointly with A.Berenstein.