Publisher Summary This chapter describes domain decomposition (DD) and multilevel methods. DD is a very natural framework in which to develop solution methods for parallel computers. Although the idea is quite old and it has been used for many years, mainly in structural mechanics, the interest in domain decomposition was renewed from the end of the eighties. This interest was motivated by the advent of parallel computers with physically distributed memories. DD has generally been used for linear systems arising from PDEs discretization. As most of the methods are closely related to partitioning the domain on which the PDE is to be solved, it is not always possible to study all DD methods from a purely algebraic point of view. Therefore, it goes back to the continuous problem and the discretization of the PDE problem. The problems are restricted that give rise to symmetric positive definite matrices although some DD methods have been developed for indefinite and non-symmetric problems.