  We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a
certain geometrically-defined category of diagonals of a regular $nm+2$-gon.
This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The
approach uses the theory of translation quivers and their corresponding mesh
categories. We also introduce the notion of the $m$th power of a translation
quiver and show how it can be used to realise the $m$-cluster category in terms
of the cluster category.
