This paper presents and discusses a wide range of solution methods for linear systems in saddle point form, with a focus on iterative methods for large and sparse problems. Saddle point problems arise in many applications in computational science and engineering, such as fluid dynamics, structural mechanics, and optimization. These systems are challenging due to their indefiniteness and poor spectral properties. The paper reviews various solution techniques, including Schur complement reduction, null space methods, coupled direct solvers, stationary iterations, Krylov subspace methods, preconditioners, and multilevel methods. It also discusses the importance of understanding the origin of the problem when choosing a preconditioner. The paper highlights the importance of the Schur complement in analyzing saddle point matrices and provides a detailed discussion of the properties of these matrices, including their invertibility, spectral properties, and conditioning. The paper also discusses the application of saddle point systems in various fields, such as incompressible flow problems, constrained optimization, and interior point methods. The paper concludes with a discussion of the challenges in solving saddle point systems and the importance of developing efficient and robust solution methods for these problems.This paper presents and discusses a wide range of solution methods for linear systems in saddle point form, with a focus on iterative methods for large and sparse problems. Saddle point problems arise in many applications in computational science and engineering, such as fluid dynamics, structural mechanics, and optimization. These systems are challenging due to their indefiniteness and poor spectral properties. The paper reviews various solution techniques, including Schur complement reduction, null space methods, coupled direct solvers, stationary iterations, Krylov subspace methods, preconditioners, and multilevel methods. It also discusses the importance of understanding the origin of the problem when choosing a preconditioner. The paper highlights the importance of the Schur complement in analyzing saddle point matrices and provides a detailed discussion of the properties of these matrices, including their invertibility, spectral properties, and conditioning. The paper also discusses the application of saddle point systems in various fields, such as incompressible flow problems, constrained optimization, and interior point methods. The paper concludes with a discussion of the challenges in solving saddle point systems and the importance of developing efficient and robust solution methods for these problems.