Nonlinear self-duality and supersymmetry

We review self-duality of nonlinear electrodynamics and its extension to se veral Abelian gauge fields coupled to scalars. We then describe self-dualit y in supersymmetric models, both N = 1 and N = 2. The self-duality equation s, which have to he satisfied by the action of any self-dual system are fou nd and solutions are discussed. One important example is the Born-Infeld ac tion. We explain why the N = 2 supersymmetric actions proposed so far are n ot the correct world-volume actions for D3 branes in d = 6.