D=2, N=2, Supersymmetric theories on Non(anti)commutative Superspace

The classical action of a two dimensional N=2 supersymmetric theory, characterized by a general K\"{a}hler potential, is written down on a non(anti)commutative superspace. The action has a power series expansion in terms of the determinant of the non(anti)commutativity parameter $C^{\alpha\beta}$. The theory is explicitly shown to preserve half of the N=2 supersymmetry, to all orders in (det C)^n. The results are further generalized to include arbitrary superpotentials as well.