The paper provides a comprehensive framework for analyzing a wide range of discontinuous Galerkin (DG) methods for second-order elliptic problems. It reviews the historical development of DG methods, from their initial introduction for hyperbolic equations in 1973 to their application to elliptic problems in recent years. The authors discuss the interior penalty (IP) methods, which were developed to enforce Dirichlet boundary conditions weakly and ensure continuity across element boundaries, and the DG methods, which were later extended to handle elliptic problems. The paper introduces the finite element spaces and the flux formulation of DG methods, showing how to derive the primal formulation from the flux formulation. It emphasizes the importance of consistent and conservative numerical fluxes in ensuring the consistency and adjoint consistency of the primal formulation. The authors present nine examples of DG methods, including the original method of Bassi and Rebay, the IP methods, and the local discontinuous Galerkin (LDG) methods, and provide their numerical fluxes and primal forms. The analysis covers the boundedness and stability of the primal bilinear form, using norms and seminorms defined on the space \( V(h) \). The paper demonstrates that for many DG methods, the primal bilinear form is bounded with respect to these norms. It also discusses the approximation properties of the finite element spaces and concludes with a unified error analysis, showing optimal error estimates in the energy norm and \( L^2 \) norm for fully stable and consistent methods. The paper highlights the importance of numerical flux choices in achieving optimal convergence rates and the challenges posed by superpenalty techniques, which can increase the condition number of the stiffness matrix.The paper provides a comprehensive framework for analyzing a wide range of discontinuous Galerkin (DG) methods for second-order elliptic problems. It reviews the historical development of DG methods, from their initial introduction for hyperbolic equations in 1973 to their application to elliptic problems in recent years. The authors discuss the interior penalty (IP) methods, which were developed to enforce Dirichlet boundary conditions weakly and ensure continuity across element boundaries, and the DG methods, which were later extended to handle elliptic problems. The paper introduces the finite element spaces and the flux formulation of DG methods, showing how to derive the primal formulation from the flux formulation. It emphasizes the importance of consistent and conservative numerical fluxes in ensuring the consistency and adjoint consistency of the primal formulation. The authors present nine examples of DG methods, including the original method of Bassi and Rebay, the IP methods, and the local discontinuous Galerkin (LDG) methods, and provide their numerical fluxes and primal forms. The analysis covers the boundedness and stability of the primal bilinear form, using norms and seminorms defined on the space \( V(h) \). The paper demonstrates that for many DG methods, the primal bilinear form is bounded with respect to these norms. It also discusses the approximation properties of the finite element spaces and concludes with a unified error analysis, showing optimal error estimates in the energy norm and \( L^2 \) norm for fully stable and consistent methods. The paper highlights the importance of numerical flux choices in achieving optimal convergence rates and the challenges posed by superpenalty techniques, which can increase the condition number of the stiffness matrix.