We have used Monte Carlo simulation and the Gibbs-Duhem integration technique to calculate temperature-composition and pressure-temperature phase diagrams for binary Lennard-Jones mixtures. We systematically explore Lennard-Jones parameter space to demonstrate how the features of these phase diagrams change as a function of diameter ratio, well-depth ratio, binary interaction parameter, and pressure. We find a number of cases in which the vapor-liquid and liquid-liquid equilibria become metastable with respect to the solid-liquid and solid-vapor equilibria.