The &nite volume element method is a discretization technique for partial di3erential equations, but in general case the coe5cient matrix of its linear system is not symmetric, even for the self-adjoint continuous problem. In this paper we develop a kind of symmetric modi&ed &nite volume element methods both for general self-adjoint elliptic and for parabolic problems on general discretization, their coe5cient matrix are symmetric. We give the optimal order energy norm error estimates. We also prove that the di3erence between the solutions of the &nite volume element method and symmetric modi&ed &nite volume element method is a high order term.