Solute transport in soils and sediments is commonly simulated with the parabolic advective-dispersive equation, or ADE. Although the solute dispersivity in this equation is regarded as a constant, it has been found to increase with the distance from the solute source. This can be explained assuming the movement of solute particles belongs to the family of Le´vy motions. A one-dimensional solute transport equation was derived for Le´vy motions using fractional derivatives to describe the dispersion. This fractional advective-dispersive equation, or FADE, has two parameters — the fractional dispersion coefficient and the order of fractional differentiation α, 0<α≤2. Scale effects are reflected by the value of α, and the fractional dispersion coefficient is independent of scale. The ADE is a special case of the FADE. Our objectives were (a) to test applicability of the FADE to field data on solute transport in soils, and (b) to develop a numerical method to solve FADE that would assure the solute mass conservation. Analytical solutions of the FADE and the ADE were successfully fitted to the data from field experiments on chloride transport in sandy loam and bromide transport in clay loam soils. A numerical method to solve the boundary problem for FADE was proposed and tested, that uses the mass-conserving flux boundary condition. The FADE is a promising model to address the scale-dependence in solute dispersion in soils and sediments.