  We show that the noncommutativity of space-time destroys the
renormalizability of the 1/N expansion of the O(N) Gross-Neveu model. A similar
statement holds for the noncommutative nonlinear sigma model. However, we show
that, up to the subleading order in 1/N expansion, the noncommutative
supersymmetric O(N) nonlinear sigma model becomes renormalizable in D=3. We
also show that dynamical mass generation is restored and there is no
catastrophic UV/IR mixing. Unlike the commutative case, we find that the
Lagrange multiplier fields, which enforce the supersymmetric constraints, are
also renormalized. For D=2 the divergence of the four point function of the
basic scalar field, which in D=3 is absent, cannot be eliminated by means of a
counterterm having the structure of a Moyal product.
