Self-diffusion coefficients and Maxwell-Stefan diffusion coefficients in liquids have been determined by the equilibrium molecular dynamics calculation of the appropriate Green-Kubo equation. Simulations of water, methanol and mixtures of water and methanol have been carried out to calculate the diffusion coefficients at 300 K. In order to study the influence of the force field on the calculated self-diffusion coefficients of the pure liquids, two different force fields for each component have been used. The Van Leeuwen/Smit force field calculated the self-diffusion of methanol accurately. The SPC/E force field gave the best, but moderate, results for water. In mixtures of water and methanol the self-diffusion coefficients of both components were more accurate at high mole fractions of methanol. This can be explained by the better performance of the methanol force field. The Maxwell-Stefan diffusion coefficients in the mixtures of methanol and water agreed fairly well with the experimental values. More accurate results can be obtained by using optimised parameters in the water force field, and by enlarging the integration time and the duration of the simulation runs.