The diffusion coefficients of ions and of uncharged solutes in aqueous solution at 25 degrees C and at infinite dilution are studied by computer simulation using the SPC/E model for water and solute-water potentials employed in previous work (Koneshan, S.; et al. J. Phys. Chem. 1998, 102, 4193-4204). The mobilities of the ions calculated from the diffusion coefficients pass through a maximum as a function of ion size, with distinct curves and maximums for positive and negative ions in qualitative agreement with experiment. We aim to understand this at a microscopic level in terms of theoretical studies of the friction coefficient zeta, which is related to diffusion coefficient by the Stokes-Einstein relation. This provides one method of calculating zeta, but it can also be obtained in principle from the random force autocorrelation function which is the starting point of molecular theories of the friction. Molecular and continuum theories divide the friction into hydrodynamic and dielectric components calculated in different ways on the basis of certain approximations that are tested in this paper. The two methods of determining zeta give consistent results in simulations of large ions or uncharged solutes but differ by a factor of nearly 1.5 for smaller ions. This is attributed to the assumption, in our simulations and in some molecular theories, that the random force autocorrelation function of a moving ion can be approximated by the total force autocorrelation function of a fixed ion but the observed trends in zeta with ion size are unchanged. Three different separations of the force autocorrelation function are studied; namely, partitioning this into (a) electrostatic and Lennard-Jones forces (b) hard and soft forces, and (c) forces arising from the first shell and more distant forces. The cross-terms are found to be significant in all cases, and the contributions of the sum of the soft term and the cross-terms, which are of opposite sign, to the total friction in the hard-soft separation, is small for all the ions (large and small). This suggests that dielectric friction calculated using this separation, with the neglect of cross-terms, is less accurate than previously supposed and the success of these theories is due to a cancellation of errors in the approximations. This is supported by recent a theoretical study of Chong and Hirata (Chong, C.; Hirata, F. J. Chem. Phys. 1998, 108, 739) which evaluates the cross-terms. A phenomenological theory due to Chen and Adelman (Chen, J. H.; Adelman, S. A. J. Chem. Phys. 1980, 72, 2819) calculates the friction in terms of effective hydrodynamic and dielectric radii for the ions (cf. solventberg picture) and predicts low dielectric friction for large ions and small strongly solvated ions. This also agrees qualitatively with our simulations but a complete molecular theory, applicable to positive and negative ions in hydrogen-bonded solvents such as water, has yet to be developed.