In this paper we introduce a family of discontinuous finite element methods for fairly general second-order elliptic equations with variable coefficients. Lower-order terms are included in these equations, so the analysis and results apply also to time-dependent equations. We first write this family of methods in a mixed formulation, and then establish their equivalent versions in a nonmixed formulation by incorporating some projection operators. Within this framework, we can recover all existing discontinuous finite element methods by changing certain terms. Stability and convergence properties are studied for these discontinuous methods; stability results and sharp error estimates are established for general boundary conditions and with reasonable assumptions. We show that when discontinuous finite element methods are defined in mixed form, they not only preserve good features of these methods, they also have some advantages over classical Galerkin discontinuous methods such as they are more stable in this form.