The Maier-Saupe theory of the nematic phase transitions, with the requirement that the molecular directors are restricted to the Cartesian axes, can be formulated as a fully-connected lattice problem, which is amenable to standard statistical-mechanics calculations. A simple three-state Maier-Saupe-Zwanzig (MSZ) model describes the weak first-order transition between uniaxial nematic and isotropic phases in liquid crystals, and provides a framework for several investigations. We describe calculations for a disordered MSZ model, which has been conceived to represent a binary mixture of rods and discs. Conflicting results in the literature about the stability of a long searched for biaxial phase are shown to be related to distinct treatments of disorder degrees of freedom. Effects of intrinsic biaxiality can be accounted for by analyzing a six-state MSZ model. We then used this model to investigate the phase diagrams of a mixture of uniaxial and intrinsically biaxial molecules, which is shown to display several nematic phases, reentrant regions, and a Landau multicritical point. We finally point out the connections of the elementary statistical models with a recent proposal of a mean-field two-tensor formalism.