A recently suggested mixture theory for the prediction of mixture properties is formulated for nonspherical molecules. The theory is capable of handling equations of state that are not conformal (do not obey the corresponding states principle). The exact composition dependence of the density expansion of the mixture equations, plus other features of the original theory, are retained. Two methods for evaluating the cross molecular parameters for nonspherical molecules are suggested. They become identical for spherical molecules, and give similar predictions for mixture properties of nonspherical molecules at high densities. The two methods predict the properties of mixtures of convex bodies and fused hard spheres within simulation uncertainty. Properties of diatomic Lennard-Jones mixtures, and ethane-carbon dioxide are predicted fairly accurately.