The recently developed equation of state (EOS) for Lennard-Jones mixtures [Y. Tang, B.C.-Y. Lu, Fluid Phase Equilibria 146 (1998) 73.] is further investigated in this work for describing phase equilibria of these mixtures. The investigation covers vapor-liquid equilibria (VLE), liquid-liquid equilibria (LLE), vapor-liquid-liquid equilibria (VLLE) and vapor-vapor equilibria (VVE) over a wide range of temperatures, pressures and molecular characteristic parameters. Results from the van der Waals one-fluid (VDW1) theory are included for comparison. The newly proposed theory performs very well for VLE and LLE and the performance is better than the VDW1 theory; but both theories yield only qualitative results for VVE. It is also found that one system should exhibit VLLE, which was not noticed in previous investigations. Results from two other perturbation theories are also compared in some cases.