  Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give
a combinatorial framework for phenomena occurring in the context of algebraic
groups. Cluster algebras also have links to a wide range of other subjects,
including the representation theory of finite dimensional algebras, as first
discovered by Marsh- Reineke-Zelevinsky. Modifying module categories over
hereditary algebras, cluster categories were introduced in work with
Buan-Marsh-Reineke-Todorov in order to "categorify" the essential ingredients
in the definition of cluster algebras in the acyclic case. They were shown to
be triangulated by Keller. Related work was done by Geiss-Leclerc-Schr\"oer
using preprojective algebras of Dynkin type. In work by many authors there have
been further developments, leading to feedback to cluster algebras, new
interesting classes of finite dimensional algebras, and the investigation of
categories of Calabi-Yau dimension $2.$
