This technical report, dedicated to Gil Strang on his 70th birthday, focuses on the numerical solution of saddle point problems, which are prevalent in computational science and engineering. The authors, Michele Benzi, Gene H. Golub, and Jörg Liesen, review a wide range of solution methods, with an emphasis on iterative techniques for large and sparse problems. The paper begins by introducing the problem statement and classification of saddle point systems, highlighting their importance in various applications such as fluid dynamics, optimization, and structural analysis. It discusses the sparsity, structure, and size of these systems, emphasizing their practical relevance. The report then delves into specific applications, including incompressible flow problems, constrained and weighted least squares, and interior point methods in constrained optimization. The properties of saddle point matrices, such as invertibility conditions and spectral properties, are explored in detail, providing theoretical foundations for solution algorithms. The report concludes with a discussion on the conditioning of these matrices and the importance of ensuring uniform invertibility in finite element contexts.This technical report, dedicated to Gil Strang on his 70th birthday, focuses on the numerical solution of saddle point problems, which are prevalent in computational science and engineering. The authors, Michele Benzi, Gene H. Golub, and Jörg Liesen, review a wide range of solution methods, with an emphasis on iterative techniques for large and sparse problems. The paper begins by introducing the problem statement and classification of saddle point systems, highlighting their importance in various applications such as fluid dynamics, optimization, and structural analysis. It discusses the sparsity, structure, and size of these systems, emphasizing their practical relevance. The report then delves into specific applications, including incompressible flow problems, constrained and weighted least squares, and interior point methods in constrained optimization. The properties of saddle point matrices, such as invertibility conditions and spectral properties, are explored in detail, providing theoretical foundations for solution algorithms. The report concludes with a discussion on the conditioning of these matrices and the importance of ensuring uniform invertibility in finite element contexts.