**Summary:** This paper by Jean Roberts and J.M. Thomas presents a comprehensive survey of mixed and hybrid finite element methods, focusing on their application to second-order elliptic problems in $ \mathbb{R}^n $, where $ n = 2, 3 $. The authors provide detailed descriptions of these methods, including convergence results and error estimates, and discuss their theoretical foundations and practical implementations. Mixed finite element methods involve the simultaneous approximation of two or more vector fields defined on the physical domain, typically the displacement field and the stress tensor. These methods are derived from the variational formulation of the problem, where the solution is sought as a pair of variables satisfying certain constraints. The Hellinger-Reissner principle is used to formulate the problem as a saddle point problem, which is then discretized using finite elements. Hybrid finite element methods, on the other hand, introduce a Lagrangian multiplier to enforce the equilibrium condition, allowing for the approximation of the stress tensor without directly solving for the displacement field. The multiplier is defined on the union of the boundaries of the elements and represents the trace of the displacement field. This approach leads to a more efficient solution process, as it avoids the need to satisfy the equilibrium condition on the entire domain. The paper also discusses the theoretical underpinnings of these methods, including the use of Sobolev spaces, trace theorems, and the properties of the spaces $ H^1(\Omega) $, $ H(\text{div}; \Omega) $, and their duals. It highlights the importance of ensuring continuity of the normal traces across element interfaces and the role of Lagrange multipliers in enforcing the equilibrium condition. The authors present several examples of mixed and hybrid methods, including primal and dual formulations, and discuss their convergence properties. They also compare these methods with conforming and equilibrium methods, emphasizing the advantages of mixed and hybrid approaches in handling complex boundary conditions and ensuring the accuracy of the solution. In conclusion, the paper provides a thorough overview of mixed and hybrid finite element methods, their mathematical foundations, and their applications to elliptic problems. It serves as a foundational reference for researchers and practitioners in the field of numerical analysis and computational mechanics.**Summary:** This paper by Jean Roberts and J.M. Thomas presents a comprehensive survey of mixed and hybrid finite element methods, focusing on their application to second-order elliptic problems in $ \mathbb{R}^n $, where $ n = 2, 3 $. The authors provide detailed descriptions of these methods, including convergence results and error estimates, and discuss their theoretical foundations and practical implementations. Mixed finite element methods involve the simultaneous approximation of two or more vector fields defined on the physical domain, typically the displacement field and the stress tensor. These methods are derived from the variational formulation of the problem, where the solution is sought as a pair of variables satisfying certain constraints. The Hellinger-Reissner principle is used to formulate the problem as a saddle point problem, which is then discretized using finite elements. Hybrid finite element methods, on the other hand, introduce a Lagrangian multiplier to enforce the equilibrium condition, allowing for the approximation of the stress tensor without directly solving for the displacement field. The multiplier is defined on the union of the boundaries of the elements and represents the trace of the displacement field. This approach leads to a more efficient solution process, as it avoids the need to satisfy the equilibrium condition on the entire domain. The paper also discusses the theoretical underpinnings of these methods, including the use of Sobolev spaces, trace theorems, and the properties of the spaces $ H^1(\Omega) $, $ H(\text{div}; \Omega) $, and their duals. It highlights the importance of ensuring continuity of the normal traces across element interfaces and the role of Lagrange multipliers in enforcing the equilibrium condition. The authors present several examples of mixed and hybrid methods, including primal and dual formulations, and discuss their convergence properties. They also compare these methods with conforming and equilibrium methods, emphasizing the advantages of mixed and hybrid approaches in handling complex boundary conditions and ensuring the accuracy of the solution. In conclusion, the paper provides a thorough overview of mixed and hybrid finite element methods, their mathematical foundations, and their applications to elliptic problems. It serves as a foundational reference for researchers and practitioners in the field of numerical analysis and computational mechanics.