We present a molecular theory for investigating the dynamics of ions in polar liquids. The theory is based on the interaction-site model for molecular liquids and on the generalized Langevin equation combined with the mode-coupling theory. The velocity autocorrelation function, diffusion and friction coefficients of ions in water at 25 degrees C and at infinite dilution are studied. The theoretical results for the velocity autocorrelation functions exhibit a gradual change from oscillatory to monotonic decay as the ion size increases. The diffusion (friction) coefficients of ions in aqueous solutions pass through a maximum (minimum) as a function of the ion size, with distinct curves and maxima (minima) for positive and negative ions. These trends are in complete accord with those of the molecular dynamics simulation results performed on the same system by Rasaiah and co-workers [J. Phys. Chem. B 102, 4193 (1998)]. It is worthwhile to mention that this is the first molecular theory that is capable of describing the difference in the dynamics of positive and negative ions in aqueous solutions. A further analysis of the friction coefficients of ions in water is presented in which the friction is decomposed into the "Stokes," dielectric and their cross terms. The Stokes and dielectric terms arise from the coupling of the ion dynamics to essentially the acoustic dynamics of the solvent via the short-range interaction, and from the coupling to the optical mode of the solvent via the long-range interaction. The most striking feature of our results is that the Stokes friction so defined does not increase monotonically with increasing ion size, but decreases when ions are very small, implying a formation of a molecular "complex" comprising the ion and its nearest neighbor solvent molecules. Interesting observations concerning the cross term are: (1) its magnitude is rather large for small ions and cannot be neglected at all, and (2) the cross term for small ions seems to cancel out the Stokes part, and consequently the total friction for small ions seems to be to a large extent determined by its dielectric component.