[@NCreview] for references. A version of Standard Model on the canonically deformed space-time was constructed in [@NCSM] and its renormalizability properties were discussed in [@NCSMRenorm]. Renormalizability of different noncommutative field theory models was discussed in [@NCRenorm].

A natural further step is modification of the superspace and introduction of non(anti)commutativity. A strong motivation for this comes from string theory. Namely, it was discovered that a noncommutative superspace can arise when a superstring moves in a constant gravitino or graviphoton background [@Seiberg], [@NACStrings]. Since that discovery there has been a lot of work on this subject and different ways of deforming superspace have been discussed. Here we mention some of them.

The authors of [@luksusy] combine SUSY with the $\kappa$-deformation of space-time, while in [@MWsusy] SUSY is combined with the canonical deformation of space-time. In [@Seiberg] a version of non(anti)commutative superspace is defined and analyzed. The anticommutation relations between fermionic coordinates are modified in the following way $$\{ \theta^\alpha \ds \theta^\beta\} =C^{\alpha\beta} , \quad \{
\bar{\theta}_{\dot\alpha} \ds \bar{\theta}_{\dot\beta}\} = \{
\theta^\alpha \ds \bar{\theta}_{\dot\alpha}\} = 0\ ,
\label{eucliddef}$$ where $C^{\alpha\beta} = C^{\beta\alpha}$ is a complex, constant symmetric matrix. This deformation is well defined only when undotted and dotted spinors are not related by 