  Using the algebraic geometric approach of Berenstein et {\it al}
(hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non
commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with
discrete torsion. We first develop a new way of getting complex $d$ mirror
Calabi-Yau hypersurfaces $H_{\Delta}^{\ast d}$ in toric manifolds $M_{\Delta
}^{\ast (d+1)}$ with a $C^{\ast r}$ action and analyze the general group of the
discrete isometries of $H_{\Delta}^{\ast d}$. Then we build a general class of
$d$ complex dimension NC mirror Calabi-Yau orbifolds where the non
commutativity parameters $\theta_{\mu \nu}$ are solved in terms of discrete
torsion and toric geometry data of $M_{\Delta}^{(d+1)}$ in which the original
Calabi-Yau hypersurfaces is embedded. Next we work out a generalization of the
NC algebra for generic $d$ dimensions NC Calabi-Yau manifolds and give various
representations depending on different choices of the Calabi-Yau toric geometry
data. We also study fractional D-branes at orbifold points. We refine and
extend the result for NC $% (T^{2}\times T^{2}\times
T^{2})/(\mathbf{{Z_{2}}\times {Z_{2})}}$ to higher dimensional torii orbifolds
in terms of Clifford algebra.
