  In this paper, we completely classify the irreducible Harish-Chandra modules
over the full toroidal Lie algebra, which is a natural higher-dimensional
analogue of the affine-Virasoro algebra. As a by-product, we also obtain the
classification of all possible irreducible Harish-Chandra modules over the
higher-dimensional Virasoro algebra. These directly generalize the well-known
result of O. Mathieu [Invent. Math. 1992] for the classical Virasoro algebra
and also the recent work of Billig-Futorny [J. Reine Angew. Math. 2016] for the
higher rank Witt algebra. In particular, we show that any irreducible
Harish-Chandra module over the higher-dimensional Virasoro algebra turns out to
be either a quotient of a module of tensor fields on a torus or a highest
weight type module up to a twist of an automorphism, as conjectured by Eswara
Rao in 2004.
