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.Comment: 24 page