A transport equation that uses fractional-order dispersion derivatives has fundamental solutions that are L¿vy's &agr;-stable densities. These densities represent plumes that spread proportional to time1/&agr;, have heavy tails, and incorporate any degree of skewness. The equation is parsimonious since the dispersion parameter is not a function of time or distance. The scaling behavior of plumes that undergo L¿vy motion is accounted for by the fractional derivative. A laboratory tracer test is described by a dispersion term of order 1.55, while the Cape Cod bromide plume is modeled by an equation of order 1.65 to 1.8. ¿ 2000 American Geophysical Union