  By Auslander's algebraic McKay correspondence, the stable category of
Cohen-Macaulay modules over a simple singularity is equivalent to the
$1$-cluster category of the path algebra of a Dynkin quiver (i.e. the orbit
category of the derived category by the action of the Auslander-Reiten
translation). In this paper we give a systematic method to construct a similar
type of triangle equivalence between the stable category of Cohen-Macaulay
modules over a Gorenstein isolated singularity $R$ and the generalized (higher)
cluster category of a finite dimensional algebra $\Lambda$. The key role is
played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra
of $R$ as well as the higher preprojective algebra of an extension of
$\Lambda$. As a byproduct, we give a triangle equivalence between the stable
category of graded Cohen-Macaulay $R$-modules and the derived category of
$\Lambda$. Our main results apply in particular to a class of cyclic quotient
singularities and to certain toric affine threefolds associated with dimer
models.
