Some aspects of deformations of supersymmetric field theories

We investigate some aspects of Moyal-Weyl deformations of superspace and their compatibility with supersymmetry. For the simplest case, when only bosonic coordinates are deformed, we consider a 4D supersymmetric field theory which is the deformation of the Wess-Zumino renormalizable theory of a chiral superfield. We then consider the deformation of a free theory of an abelian vector multipier, which is a non-commutative version of the rank 1 Yang-Mills theory. We finally give the supersymmetric version of the alpha-prime approaches 0 limit of the Born-Infeld action with a B-field turned on, which is believed to be related to the non-commutativc U(1) gauge theory. (Author)