The Research in this area developed with amazing speed after the introduction of cluster algebras around 2001 by Sergey Fomin and Andrei Zelevinsky and has attracted a variety of first rate mathematicians throughout the world, for instance Alexander Goncharov, Bernhard Keller, Maxim Kontsevich, Bernard Leclerc, Idun Reiten and Claus Michael Ringel, most of them being ICM speakers.
A good way to get an overview of the intense activities related to
cluster algebras is Sergey Fomin's cluster algebras portal:
http://www.math.lsa.umich.edu/~fomin/cluster.html
see also section below for some discussion of the
impact of cluster algebras.