This is an expository paper on the localization of simply connected spaces. The form of this theory is due to Dror Farjoun and to Bousfield. It includes in one treatment the classical localization of inverting primes and that of p−completion. This paper presents these localizations in a geometric form for spaces and also in a closely related algebraic form for abelian groups. In addition, it includes the exotic localizations related to Miller’s theorem. These exotic localizations give another proof of Serre’s theorem that simply connected finite complexes are either contractible or have infinitely many nonzero homotopy groups. We also give a proof of Serre’s conjecture that the homotopy groups of these spaces are either all zero or have infinitely many nonzero torsion components. This paper is based on lectures which were given in Braga, Portugal and in Lausanne, Switzerland. I would like to thank Lućıa Fernández-Suárez and Kathryn Hess for their support.