This paper, authored by Jean Roberts and J.M. Thomas, provides an expository survey of mixed and hybrid finite element methods. It serves as a basic reference on the subject, detailing the methods, including convergence results and error estimates, for a second-order elliptic problem in \(\mathbb{R}^n\) with \(n = 2, 3\). The authors introduce the concepts of mixed and hybrid methods, which are extensions of conforming and equilibrium finite element methods, respectively. They discuss the use of Lagrangian multipliers to transform constrained minimization problems into saddle-point problems, leading to the development of mixed methods. Hybrid methods, on the other hand, relax the regularity requirement on the space where the stress is approximated while retaining the equilibrium requirement. The paper covers various types of finite element interpolation for scalar and vectorial functions, primal and dual mixed methods, primal and dual hybrid methods, and the hybridization of equilibrium methods. It also includes extensions and variations of the theory, providing a comprehensive overview of the field.This paper, authored by Jean Roberts and J.M. Thomas, provides an expository survey of mixed and hybrid finite element methods. It serves as a basic reference on the subject, detailing the methods, including convergence results and error estimates, for a second-order elliptic problem in \(\mathbb{R}^n\) with \(n = 2, 3\). The authors introduce the concepts of mixed and hybrid methods, which are extensions of conforming and equilibrium finite element methods, respectively. They discuss the use of Lagrangian multipliers to transform constrained minimization problems into saddle-point problems, leading to the development of mixed methods. Hybrid methods, on the other hand, relax the regularity requirement on the space where the stress is approximated while retaining the equilibrium requirement. The paper covers various types of finite element interpolation for scalar and vectorial functions, primal and dual mixed methods, primal and dual hybrid methods, and the hybridization of equilibrium methods. It also includes extensions and variations of the theory, providing a comprehensive overview of the field.