The revised valence-bond (VB) theory of previous papers is applied in non-empirical calculation of the lower $\pi $-electron levels of cyclobutadiene and benzene. For the first of these molecules, complete sets of all non-polar and polar, singlet and triplet structures are employed; for the second, 89 singlet and 69 triplet structures were selected from the complete sets of 175 singlet and 189 triplet structures. The calculations, which follow unfamiliar lines but which proved not too heavy, are outlined in some detail. The results for cyclobutadiene agree with those of Craig, who employed a complete MO basis; but previous calculations on benzene are surpassed in accuracy, except in the case of the ground state which is apparently well represented by a few MO configurations. The primary aim of the present work was, however, simply to exploit the valence-bond approach as a practicable alternative to the MO method, with exactly similar 'non-empirical' potentialities. Energy levels, wave functions and bond orders were calculated in a variety of approximations, numerical results converging slowly to their final values as configurations were added, so that the function of different types of structure should be revealed: and 'higher' structures-those, for instance, which are doubly-polar-were found to have an unexpected importance. This supports the view that calculations making any pretence of being non-empirical cannot easily be extended to systems which explicitly involve more than a few electrons. The orthodox VB approach is re-examined, within this rigorous framework, and is found to have less intrinsic value than might have been hoped; the shortcomings of the conventional empirical theory are further revealed. Semi-empirical developments are briefly investigated. The revised VB theory contains only a few numerically large parameters: these are, in the first place, 'resonance' integrals (of the one-electron type encountered in MO theory) and, secondly, the energies of the various polar configurations relative to the non-polar; by adjusting the latter quantities empirically it is possible partly to overcome one of the principal defects of the non-empirical theory-namely, its use of the Huckel approximation in which changes in the $\sigma $-bonded framework, accompanying $\pi $-electron 'polarizations', are ignored. It seems likely that considerable progress can be made along these lines.