In this paper we present a reliable algorithm, the homotopy perturbation method, to construct numerical solutions of the space-time fractional advection-dispersion equation in the form of a rapidly convergent series with easily computable components. Fractional advection-dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium. The fractional derivatives are described in the Caputo sense. Some examples are given. Numerical results show that the homotopy perturbation method is easy to implement and accurate when applied to space-time fractional advection-dispersion equations. (C) 2009 Elsevier Ltd. All rights reserved.