  We investigate the most general non(anti)commutative geometry in N=1
four-dimensional superspace, invariant under the classical (i.e., undeformed)
supertranslation group. We find that a nontrivial non(anti)commutative
superspace geometry compatible with supertranslations exists with
non(anti)commutation parameters which may depend on the spinorial coordinates.
The algebra is in general nonassociative. Imposing associativity introduces
additional constraints which however allow for nontrivial commutation relations
involving fermionic coordinates. We obtain explicitly the first three terms of
a series expansion in the deformation parameter for a possible associative
*-product. We also consider the case of N=2 euclidean superspace where the
different conjugation relations among spinorial coordinates allow for a more
general supergeometry.
