There is no comprehensive model for the dynamics of cellular membranes. Even mechanisms of basic dynamic processes, such as lateral diffusion of lipids, are poorly understood. Our atomic-scale molecular dynamics simulations support a novel, concerted mechanism for lipid diffusion. We find that a lipid and its nearest neighbors move in unison, forming loosely defined clusters. What is more, the motions of lipids are correlated over tens of nanometers:  the lateral displacements of lipids in a given monolayer produce striking two-dimensional flow patterns. These flow patterns should have wide implications, affecting, for example, the formation of membrane domains, protein functionality, and action of lipases and drugs on membranes.