In this paper the free volume theory of diffusion is extended to multicomponent mixtures. The free volume is taken to be accessible for any component according to its surface. fraction. The resulting equations predict multicomponent (Maxwell-Stefan) diffusivities in simple liquid mixtures from pure component data such as molar masses, densities and viscosities. They can also be used together with an equation of state. For simple Liquid mixtures, the results agree closely with experiment. For viscous liquids, rubbery polymers and glasses, orders of magnitude and the main trends of diffusivities of small permeants are predicted correctly. The equations follow many of the empirical rules known for diffusion coefficients. In non-viscous liquids they almost coincide with an empirical modification of the Einstein-Stokes equation. They predict an Arrhenius type of temperature dependence with correct activation energies. In mixtures with similar components, they predict that MS-diffusivities should be Linear functions of composition (the Darken rule). In mixtures with larger (but not-too-large) differences between the components, the MS-diffusivities are logarithmic functions of composition (the Vignes rule). In viscous mixtures and polymers, diffusivities vary sharply (in a roughly exponential manner) with the concentration of the viscous component. The theory is not perfect, It fails for mixtures of molecules differing greatly in size or chemical structure. It can only be made to work with water and polymers with a few dubious assumptions.