Molecular-dynamics computer simulation has been used to calculate the self-diffusion coefficient of a Lennard-Jones fluid in the interval of reduced temperatures T* = 0.35-2.0, through mean-squared particles displacements and the velocity autocorrelation function. Calculations have been performed for systems in stable and metastable states up to the boundary of spontaneous nucleation in a model of a fluid consisting of 2048 particles. The values of the self-diffusion coefficient on the spinodals of a superheated liquid and supersaturated vapor have been determined. It has been established that in the region of supercooled states of the liquid phase on an isotherm the logarithm of the self-diffusion coefficient is a linear function of the pressure. (C) 2010 Elsevier B.V. All rights reserved.