The mixed quantum classical theory, MQCT, for inelastic scattering of two molecules is developed, in which the internal (rotational, vibrational) motion of both collision partners is treated with quantum mechanics, and the molecule-molecule scattering (translational motion) is described by classical trajectories. The resultant MQCT formalism includes a system of coupled differential equations for quantum probability amplitudes, and the classical equations of motion in the mean-field potential. Numerical tests of this theory are carried out for several most important rotational state-to-state transitions in the N-2 + H-2 system, in a broad range of collision energies. Besides scattering resonances (at low collision energies) excellent agreement with full-quantum results is obtained, including the excitation thresholds, the maxima of cross sections, and even some smaller features, such as slight oscillations of energy dependencies. Most importantly, at higher energies the results of MQCT are nearly identical to the full quantum results, which makes this approach a good alternative to the full-quantum calculations that become computationally expensive at higher collision energies and for heavier collision partners. Extensions of this theory to include vibrational transitions or general asymmetric-top rotor (polyatomic) molecules are relatively straightforward.