This paper provides a unified analysis of a large class of discontinuous Galerkin (DG) methods for second-order elliptic problems. It aims to understand and compare most of the DG methods proposed over the past three decades for elliptic problems. The authors introduce a framework that allows for the analysis of these methods by considering a first-order system formulation of the elliptic problem. They define the DG methods in terms of a flux formulation, where the numerical fluxes are approximations of the true fluxes on the boundaries of elements. The numerical fluxes play a crucial role in the stability and accuracy of the method, and their choice is discussed in detail. The paper then introduces the finite element spaces associated with the triangulation of the domain and defines the primal formulation of the DG methods by eliminating the auxiliary variable σ_h. The authors relate the properties of consistency and conservativity of the numerical fluxes to the properties of consistency and adjoint consistency of the primal formulation. They analyze several examples of DG methods, including the original DG method of Bassi and Rebay, the interior penalty (IP) methods, and the local discontinuous Galerkin (LDG) methods. The authors perform a unified error analysis of these methods, showing that optimal error estimates can be achieved in the energy norm for fully stable and consistent methods. They also discuss the effects of relaxing the consistency and stability conditions, showing that optimal error estimates can still be obtained by using large penalty weights, although this may increase the condition number of the stiffness matrix. The paper concludes with a summary of the main features of the various methods, a discussion of possible extensions, and an overview of ongoing work and open problems in the field of DG methods for elliptic problems. The authors emphasize the importance of the numerical fluxes in the stability and accuracy of the DG methods and highlight the role of the primal formulation in the analysis of these methods. The paper provides a comprehensive overview of the development and analysis of DG methods for elliptic problems, and it serves as a valuable resource for researchers and practitioners in the field.This paper provides a unified analysis of a large class of discontinuous Galerkin (DG) methods for second-order elliptic problems. It aims to understand and compare most of the DG methods proposed over the past three decades for elliptic problems. The authors introduce a framework that allows for the analysis of these methods by considering a first-order system formulation of the elliptic problem. They define the DG methods in terms of a flux formulation, where the numerical fluxes are approximations of the true fluxes on the boundaries of elements. The numerical fluxes play a crucial role in the stability and accuracy of the method, and their choice is discussed in detail. The paper then introduces the finite element spaces associated with the triangulation of the domain and defines the primal formulation of the DG methods by eliminating the auxiliary variable σ_h. The authors relate the properties of consistency and conservativity of the numerical fluxes to the properties of consistency and adjoint consistency of the primal formulation. They analyze several examples of DG methods, including the original DG method of Bassi and Rebay, the interior penalty (IP) methods, and the local discontinuous Galerkin (LDG) methods. The authors perform a unified error analysis of these methods, showing that optimal error estimates can be achieved in the energy norm for fully stable and consistent methods. They also discuss the effects of relaxing the consistency and stability conditions, showing that optimal error estimates can still be obtained by using large penalty weights, although this may increase the condition number of the stiffness matrix. The paper concludes with a summary of the main features of the various methods, a discussion of possible extensions, and an overview of ongoing work and open problems in the field of DG methods for elliptic problems. The authors emphasize the importance of the numerical fluxes in the stability and accuracy of the DG methods and highlight the role of the primal formulation in the analysis of these methods. The paper provides a comprehensive overview of the development and analysis of DG methods for elliptic problems, and it serves as a valuable resource for researchers and practitioners in the field.