Supersymmetry, a Biased Review

This set of lectures contain a brief review of some basic supersymmetry and its representations, with emphasis on superspace and superfields. Starting from the Poincaré group, the supersymmetric extensions allowed by the Coleman-Mandula theorem and its generalisation to superalgebras, the Haag, Lopuszanski and Sohnius theorem, are discussed. Minkowski space is introduced as a quotient space and Superspace is presented as a direct generalization of this. The focus is then shifted from a general presentation to the relation between supersymmetry and complex geometry as manifested in the possible target space geometries for N=1 and N=2 supersymmetric nonlinear sigma models in four dimensions. Gauging of isometries in nonlinear sigma models is discussed for these cases, and the quotient construction is described.