  We deform the standard four dimensional $\N=1$ superspace by making the odd
coordinates $\theta$ not anticommuting, but satisfying a Clifford algebra.
Consistency determines the other commutation relations of the coordinates. In
particular, the ordinary spacetime coordinates $x$ cannot commute. We study
chiral superfields and vector superfields and their interactions. As in
ordinary noncommutative field theory, a change of variables allows us to
express the gauge interactions in terms of component fields which are subject
to standard gauge transformation laws. Unlike ordinary noncommutative field
theories, the change of the Lagrangian is a polynomial in the deformation
parameter. Despite the deformation, the noncommutative theories still have an
antichiral ring with all its usual properties. We show how these theories with
precisely this deformation arise in string theory in a graviphoton background.
