Vapor-liquid, vapor-solid, liquid-liquid, and liquid-solid coexistence lines are calculated for binary mixtures of Lennard-Jones spheres with diameter ratio sigma (11)/sigma (12) = 0.45, well-depth ratio epsilon (11)/epsilon (22) = 0.45, and binary interaction parameters delta (12) = 1.0, 0.9, and 0.75, using Monte Carlo simulation and the Gibbs-Duhem integration technique. These calculations allow us to construct complete phase diagrams, i.e. showing equilibrium between vapor, liquid, and solid phases. For the mixture with delta (12) = 1, we find a completely miscible vapor-liquid coexistence region with a eutectic solid-liquid coexistence region. These two regions are separated by a completely miscible liquid phase. For the mixtures with delta (12) < 1, we find that the vapor-liquid and solid-liquid coexistence regions interfere. This interference results in a vapor-solid coexistence region bounded above and below by solid-liquid-vapor coexistence lines. We also find that the mixtures with delta (12) < 1 have a region of liquid-liquid immiscibility that is metastable with respect to the solid-fluid phase equilibria.