In this paper we described some of the problems that lead to saddle point systems with special emphasis on incompressible flow problems and optimization problems. In order to construct efficient solution algorithms, it was necessary to understand the properties of matrices that occur as a result of discretization of a continuous problems. The most important conclusion of this theory is that matrices from saddle point systems have very poor spectral properties due to their vicinity to singular matrices. Nevertheless, using Schur complement decomposition and other techniques, it is possible to overcome certain obstacles and in combination with preconditioning construct a viable solution algorithms for this ill-conditioned systems. The first effective method developed for solving a certain class of saddle point systems is Uzawa method. Later, many methods were developed that are primary based on Krylov subspace iterations with the most prominent of them GMRES method. Unfortunately, due to ill-conditioned nature of saddle point systems, some sort of preconditioning is almost always required. Since the theory of determining a precondition matrix is very complex with many unanswered questions, the numerical examples in this paper are a bit simpler.