Researchers from many fields have shown that a natural extension of the classical advection-dispersion equation (ADE) uses fractional-order derivatives. A fractional ADE has been shown to agree with data from a number of tracer tests that defy other modeling approaches. However, a number of unanswered questions remain. We will test and extend the theory through a series of numerical and analytic studies. Among our tasks is the creation of detailed K fields with new and existing generators that exploit power-law distributions. We will compare the resulting statistics of velocity and particle transport to those predicted by the fractional theory. We will supplement the numerical work with extended analytic expressions for heavy-tailed motions to see the effect of temporal correlation on the transport equation. We will also develop and test finite-element and random walk numerical models of a fractional ADE and generalize our work in 3-D to allow for different types of motion in all directions. These extensions will be tested against the conservative solute plumes at the MADE and Cape Cod aquifers. We will also rederive the governing equations in a more straightforward Eulerian setting via fractional-order Taylor series expansions and relate parameters to measurable aquifer properties. Finally, since the diverging variance of Levy motion is somewhat problematic, we will develop alternate measures for fractal random functions and relate these to our observations.