The symmetric positive definite system arising from a finite element discretization of an elliptic boundary value problem can be solved efficiently using the preconditioned conjugate gradient method (cf. (Saad 1996)). In this chapter we discuss the class of additive Schwarz preconditioners, which has built-in parallelism and is particularly suitable for implementation on parallel computers. Many well-known preconditioners are included in this class, for example the hierarchical basis and BPX multilevel preconditioners, the two-level additive Schwarz overlapping domain decomposition pre-conditioner, and the BPS and Neumann-Neumann nonoverlapping domain decomposition preconditioners.