We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the L-p norm, 2 less than or equal to p less than or equal to infinity, are derived. We also show second-order convergence in the L-p norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the "supercloseness" results in Chou and Li [Math Comp 69(229) (2000), 103-120] to the L-p based spaces, duality arguments, and the discrete Green's function method. (C) 2003 Wiley Periodicals, Inc.