Representations of extended supersymmetry

Recent results of the quest for off-shell formulations of supersymmetric theories are presented. T technique of dimensional reduction by Legendre transformation is described which allows construction of off-shell formulations of all supersymmetric theories in few space-time dimensions. Some representations of N=2 supersymmetry are discussed and the general programme of finding superspace constraints to discover the necessary set of constraints on the N=2 Yang-Mills field strengths applied. The further constraints which are implied by the Yang-Mills Bianchi-identities are evaluated. The consequences of these constraints on the component field content of the N=2 Yang-Mills theory in ordinary space are shown. The Wess-Zumino model is obtained and a theory which has the same content as the N=2 hypermultiplet discussed, with special reference to the N=2 abelian gauge theory. The implications for the maximally extended supersymmetric Yang-Mills theory are discussed. (U.K.)