This book offers a new approach towards the introduction and analysis of multigrid methods from an algebraic point of view. This approach is independent of the traditional, geometric approach, which is based on rediscretizing the original PDE. Instead, it uses only the algebraic properties of the original linear system to define, analyze, and apply the multigrid iterative method. This way, multigrid methods are well embedded in the family of iterative methods for the numerical solution of large, sparse linear systems. Indeed, as is shown below, multigrid methods can actually be viewed as special cases of domain-decomposition methods.