This thesis studies numerical solutions of large sparse linear systems. We focus on Gauss method, iterative conjugate gradient and its variants, and preconditioned conjugate gradient methods. We describe the methods, the corresponding algorithms, and their costs and convergence rates in order to allow a motivated methods choice. We limit our study to large sparse linear systems resulting from finite elements discretization of boundary value problems. The first chapter is dedicated to a general introduction and preliminaries on the subject. The second chapter introduces finite element systems and benchmark model. Sparse systems and Gauss method are the subjects of the third chapter. We study conjugate gradient method in the fourth chapter, then preconditioned conjugate gradient methods in the fifth chapter. We compare numerically all presented methods in the sixth chapter. Then, we summarize and present some conclusions and future work in the last chapter. All MATLAB codes are presented in the last pages of the thesis..