Vapor-liquid, vapor-solid and liquid-solid coexistence lines are calculated for binary mixtures of Lennard-Jones spheres using Monte Carlo simulation and the Gibbs-Duhem integration technique. Complete phase diagrams showing equilibrium between vapor; liquid and solid phases are constructed for binary Lennard-Jones mixtures with diameter ratios ranging from sigma (11)/sigma (22) = 0.85-1.0 and attractive well-depth ratios ranging from epsilon (11)/epsilon (22) = 0.625 - 1.6, at a reduced pressure P* = P sigma (3/)(11)epsilon (11) = 0.002. The Lorentz-Berthelot combining rules are used to calculate the cross-species interaction parameters sigma (12) and epsilon (12) The variation in the shape of the complete phase diagrams change as a function of the diameter ratio sigma (11)/sigma (12), and well-depth ratio epsilon (11)/epsilon (12) is systematically explored. The phase diagrams found here resemble those Sound experimentally for argon-methane, iodine-sulfur; water-sodium chloride, wave-silver nitrate, water-potassium nitrate, and p-dichlorobenzene-p-dibromobenzene.