The theory of Abelian and non-Abelian magnetic monopoles is reviewed with special focus on the exact integrability properties of such systems. The limit of vanishing Higgs potential (Prasad-Sommerfield limit) is analyzed in detail. At the classical level, the construction of all static multimonopole solutions is presented, with emphasis on the explicit axially symmetric states. At the semiclassical level, the problems of small fluctuations, bosonic and fermionic zero modes and the construction of static propagators are discussed. Finally we consider the possibility of embedding monopoles in supersymmetric theories in order to obtain models with stronger convergence properties and possibly full quantum mechanical integrability.