  This article proposes and analyzes the generalized weak Galerkin ({\rm g}WG)
finite element method for the second order elliptic problem. A generalized
discrete weak gradient operator is introduced in the weak Galerkin framework so
that the {\rm g}WG methods would not only allow arbitrary combinations of
piecewise polynomials defined in the interior and on the boundary of each local
finite element, but also work on general polytopal partitions. Error estimates
are established for the corresponding numerical functions in the energy norm
and the usual $L^2$ norm. A series of numerical experiments are presented to
demonstrate the performance of the newly proposed {\rm g}WG method.
