The ability to measure biomolecular dynamics within cells and tissues is very important to understand fundamental physiological processes including cell adhesion, signalling, movement, division or metabolism. Usually, such information is obtained using particle tracking methods or single point fluctuation spectroscopy. We show that image mean square displacement analysis, applied to single plane illumination microscopy data, is a faster and more efficient way of unravelling rapid, three-dimensional molecular transport and interaction within living cells. From a stack of camera images recorded in seconds, the type of dynamics such as free diffusion, flow or binding can be identified and quantified without being limited by current camera frame rates. Also, light exposure levels are very low and the image mean square displacement method does not require calibration of the microscope point spread function. To demonstrate the advantages of our approach, we quantified the dynamics of several different proteins in the cyto- and nucleoplasm of living cells. For example, from a single measurement, we were able to determine the diffusion coefficient of free clathrin molecules as well as the transport velocity of clathrin-coated vesicles involved in endocytosis. Used in conjunction with dual view detection, we further show how protein-protein interactions can be quantified.