We demonstrate that hydrodynamics and fluctuations affect diffusion in liquids in crucial ways, for both molecular diffusion (fluid mixtures), and colloidal suspensions. We study diffusive mixing in the presence of thermal fluctuations when the Schmidt number is large. We obtain a closed equation for the concentration which is amenable to efficient numerical solution. This equation captures both Fick's law for the ensemble-averaged mean and also the long-range correlated giant fluctuations in individual realizations of the mixing process. These giant fluctuations, observed in experiments, are shown to be the result of the long-ranged hydrodynamic correlations among the diffusing particles. Through a combination of Eulerian and Lagrangian numerical experiments we demonstrate that mass transport in liquids can be modeled at all scales, from the microscopic to the macroscopic, not as irreversible Fickian diffusion, but rather, as reversible random advection by thermal velocity fluctuations. Our model gives effective dissipation with a diffusion coefficient that is not a material constant as its value depends on the scale of observation. Our work reveals somewhat unexpected connections between flows at small scales, dominated by thermal fluctuations, and flows at large scales, dominated by turbulent fluctuations. This is joint work with Thomas Fai and Eric Vanden-Eijnden.