  In this paper, we use subword complexes to provide a uniform approach to
finite type cluster complexes and multi-associahedra. We introduce, for any
finite Coxeter group and any nonnegative integer k, a spherical subword complex
called multi-cluster complex. For k=1, we show that this subword complex is
isomorphic to the cluster complex of the given type. We show that multi-cluster
complexes of types A and B coincide with known simplicial complexes, namely
with the simplicial complexes of multi-triangulations and centrally symmetric
multi-triangulations respectively. Furthermore, we show that the multi-cluster
complex is universal in the sense that every spherical subword complex can be
realized as a link of a face of the multi-cluster complex.
