Phase diagrams and critical constants for the long-range corrected, the truncated, and the truncated and shifted Lennard-Jones fluids with various values of the potential cutoff were computed from molecular simulations. Critical parameters were obtained from mixed-field finite-size scaling analysis. Multiple histogram reweighting was used to compute the phase envelop at temperatures well below criticality. For the long-range corrected fluid, the coexistence curve is systematically shifted to higher chemical potentials for a cutoff of 5.0 sigma compared with that for a cutoff of 2.5 sigma. The difference in the critical temperature for the truncated and truncated and shifted potentials decreases from 10 to 3.6% as the cutoff increases from 2.5 sigma to 3.5 sigma. The critical temperature for the long-range corrected fluid is about 1.4% larger than that for the truncated fluid with a cutoff of 5.0 sigma. The average absolute deviations of the coexistence densities between the truncated and long-range corrected fluid with r(c) = 5.0 sigma are about 0.8 and 1% for the vapor and liquid branches, respectively. This indicates that the truncated Lennard-Jones fluid with a cutoff of 5.0 sigma is a reasonable quantitative approximation to the full Lennard-Jones fluid.