Z(N) x Z(M) orientifolds with and without discrete torsion

We discuss compact four-dimensional Z(N) X Z(M) type-IIB orientifolds. We t ake a systematic approach to classify the possible models and construct the m explicitely. The supersymmetric orientifolds of this type have already be en constructed some time ago. We find that there exist several consistent o rientifolds for each of the discrete groups Z(2) X Z(2), Z(2) X Z(4), Z(4) X Z(4), Z(2) X Z(6), Z(2) X Z(6)' and Z(6) X Z(6) if anti-D5-branes are int roduced. Supersymmetry is broken by the open strings ending on antibranes. The rank of the gauge group is reduced by a factor two if the underlying or bifold space has discrete torsion.