Extensive molecular-dynamics, MD, simulations of Lennard-Jones, LJ, and soft-sphere, SS, fluids have been made. The LJ state points agree excellently with the predictions of a recent equation of state for this fluid. Self-diffusion coefficients obtained from the LJ mean-square displacements at equilibrium are fitted to a simple analytic expression, involving temperature and density, which has a density and temperature range of wider applicability than that of Levesque and Verlet. The corresponding shear viscosities in the limit of zero shear rate have been obtained by a new non-equilibrium MD technique, which involves orthogonal longitudinal distortions (eliminating pure expansion or compression terms). The LJ shear rigidity moduli are fitted to better than 1 % by a simple analytic expression. A similar relationship for the shear viscosities is less satisfactory but this merely reflects the greater uncertainty in this collective transport property. The Stokes–Einstein relationship using slip boundary conditions gives an effective ‘flow unit’ diameter which decreases from several molecular diameters to one as the density increases. This suggests that the motion between a molecule and those in its first coordination shell is more cooperative at moderate densities resulting in greater coupling between molecular trajectories. Support for this comes from the direct evaluation of the friction coefficient by non-equilibrium MD, pair radial and pair fluctuation correlation functions. The Verlet algorithm was used in these calculations. The more accurate Toxvaerd algorithm is shown not to improve noticeably the accuracy of the systematic component of single (and hence possibly collective) particle motion, as measured by velocity and force autocorrelation functions.