A cluster algebra, which was introduced by Fomin and Zelevinsky, is a commutative algebra with a family of distinguished generators (the cluster variables) grouped into overlapping subsets (the clusters) which are constructed by mutations. Cluster algebras have found applications and connections in a wide range of subjects including quiver representation theory, Poisson geometry, Teichm\"uller theory, tropical geometry and so forth. A quiver Grassmannian is a projective variety parametrizing subrepresentations of a quiver representation with a given dimension vector. Quiver Grassmannians include the usual Grassmannians and flag varieties as very special cases. After introducing cluster algebras and quiver Grassmannians, we show how the Euler characteristics of quiver Grassmannians can be explicitly computed by using cluster algebras. |