of a cluster algebra. This structure should serve as an algebraic framework for the study of “dual canonical bases” in these coordinate rings and their $q$-deformations.”*

In this survey we shall review the results of a series of papers [@GLS2; @GLSUni1; @GLSKM; @GLSquantum], in which we have implemented (part of) this program for certain varieties associated with elements $w$ of the Weyl group of a Kac-Moody group, namely, the unipotent subgroups $N(w)$ and the unipotent cells $N^w$.

Cluster algebras {#sect2}
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We start with a quick (and very incomplete) introduction to the theory of cluster algebras. For more developed accounts we refer to [@FominSurvey; @FZSurvey; @GSW; @Keller].

Definitions and basic results
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Consider the field of rational functions $\F=\Q(x_1,\ldots,x_n)$. A *seed* in $\F$ is a pair $\Si=(\y,Q)$, where $\y=(y_1,\ldots,y_n)$ is a free generating set of $\F$, and $Q$ is a quiver (an oriented graph) with vertices labelled by $\{1,\ldots,n\}$. We assume that $Q$ has neither loops nor 2-cycles. For $k=1,\ldots,n$, one defines a new seed $\mu_k(\Si)=(\mu_k(\y),\mu_k(Q))$ as follows. First $\mu_k(y_i) = y_i$ for $i\not = k$, and $$\label{FZmutation_form}
\mu_k(y_k)= \frac{\prod_{i\to k}y_i + \prod_{k\to j} y_j}{y_k},$$ where the first (second) product in the right hand side is over all arrows of 