Some aspects of deformations of supersymmetric field theories

We investigate some aspects of Moyal-Weyl deformations of superspace and th eir compatibility with supersymmetry. For the simplest case, when only boso nic coordinates are deformed, we consider a four-dimensional supersymmetric field theory which is the deformation of the Wess-Zumino renormalizable th eory of a chiral superfield. We then consider the deformation of a free the ory of an abelian vector multiplet, which is a non-commutative version of t he rank 1 Yang-Mills theory. We finally give the supersymmetric version of the alpha' bar right arrow 0 limit of the Born-Infeld action with a B-field turned on, which is believed to be related to the non-commutative U(1) gau ge theory.