The concept of the representative elementary volume (REV) is often associated with the notion of hydrodynamic dispersion and Fickian transport. However, it has been frequently observed experimentally and in numerical pore scale simulations that transport is non-Fickian and cannot be characterized by hydrodynamic dispersion. Does this mean that the concept of the REV is invalid? We investigate this question by a comparative analysis of the advective mechanisms of Fickian and non-Fickian dispersion and their representation in large scale transport models. Specifically, we focus on the microscopic foundations for the modeling of pore scale fluctuations of Lagrangian velocity in terms of Brownian dynamics (hydrodynamic dispersion) and in terms of continuous time random walks, which account for non-Fickian transport through broad distributions of advection times. We find that both approaches require the existence of an REV that, however, is defined in terms of the representativeness of Eulerian flow properties. This is in contrast to classical definitions in terms of medium properties such as porosity, for example. Keywords Representative elementary volume • upscaling • anomalous dispersion • continuous time random walks • velocity statistics