The paper considers the problem of computing the volume of a polyhedron in n dimensions given either as a convex hull of its vertices or as a set of linear inequalities (both representations use only rational numbers). It is shown that this problem is {\#}P-hard for both representations, i.e., it is at least as hard as computing the permanent of a matrix. On the other hand, in the vertex-based case the {\#}P-easiness of the volume problem is shown, i.e., one can compute the volume in polynomial time using a {\#}P-complete oracle. Note that the volume problem is not in class {\#}P. As for the facet-based case, the {\#}P-easiness is proven of computing V such that \(| V-Vol| <\epsilon\) where Vol is the actual volume. Some other results clarifying the volume problem complexity further are also given. See also \textit{L. G. Khachiyan}, Usp. Mat. Nauk 44, No.3(267), 179-180 (1989).