  We describe a new class of supersymmetric orientifolds which combine the
world-sheet parity transformation with a complex conjugation in the compact
directions. As an example, we investigate in detail the orientifold of the Z_3
toroidal orbifold in six and four dimensions. We demonstrate how the solution
to the tadpole cancellation conditions, the resulting gauge groups and the
massless spectra depend on the choice of the complex structures on the tori,
giving rise to a variety of inequivalent models. We also summarize the results
for the orientifolds of the Z_4, Z_6 and Z_6' orbifolds in four and six
dimensions.
