  We introduce (n+1)-preprojective algebras of algebras of global dimension n.
We show that if an algebra is n-representation-finite then its
(n+1)-preprojective algebra is self-injective. In this situation, we show that
the stable module category of the (n+1)-preprojective algebra is
(n+1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category
of the stable n-Auslander algebra of the original algebra. In particular this
stable category contains an (n+1)-cluster tilting object. We show that even if
the (n+1)-preprojective algebra is not self-injective, under certain
assumptions (which are always satisfied for n \in {1,2}) the results above
still hold for the stable category of Cohen-Macaulay modules.
