  Let $G$ be a rank $n$ additive subgroup of $\bC$ and $\Vir[G]$ the
corresponding Virasoro algebra of rank $n$. In the present paper, irreducible
weight modules with finite dimensional weight spaces over $\Vir[G]$ are
completely determined. There are two different classes of them. One class
consists of simple modules of intermediate series whose weight spaces are all
1-dimensional. The other is constructed by using intermediate series modules
over a Virasoro subalgebra of rank $n-1$. The classification of such modules
over the classical Virasoro algebra was obtained by O. Mathieu in 1992 using a
completely different approach.
