THERE is currently no unifying quantitative description of atomic diffusion in condensed matter, Analytic expressions have been obtained for the transport coefficients of an idealized dense fluid of hard spheres(1,2), but their generalization to the rich variety of atomic structures in real condensed systems remains a challenge, Here I present evidence from molecular dynamics simulations that a universal relationship exists between the structure and the equilibrium rate of atomic diffusion in liquids and solids. I find that the diffusion coefficient, reduced to a dimensionless form by scaling by the atomic collision frequency and the atomic diameter, is uniquely defined by the excess entropy, a measure of the number of accessible configurations of the system. A scaling law relating these two quantities holds well for simple liquids, and also remains applicable to atomic transport in a quasicrystal and to silver-ion diffusion in the solid-state ionic conductor alpha-AgI. This makes it possible to estimate diffusion coefficients directly from diffraction measurements of an equilibrium structural characteristic, namely the radial distribution function of the diffusing species.