We consider the application of the cell theory to single component and binary Lennard-Jones solids. We calculate solid phase properties and solid-fluid equilibrium using the cell theory for the solid phase and an equation of state for the fluid phase. In the single component case the thermodynamic properties as well as the solid-fluid phase diagram predicted by the theory are in quite good agreement with Monte Carlo simulation results. The introduction of correlations between the motions of nearest neighbor particles into the cell theory in a fashion suggested by Barker significantly improves the agreement. For binary Lennard-Jones 12-6 mixtures the predictions of the theory are compared the experimental data for mixtures forming substitutionally disordered solid solutions involving argon, krypton and methane. The theory correctly predicts the form of the phase diagram but the quantitative predictions are quite sensitive to the choice of potential parameters. The shape of the phase diagram is similar to that for a hard sphere mixture with the same diameter ratio.