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We introduce a stochastic process called a Lévy walk which is a random walk with a nonlocal memory coupled in space and in time in a scaling fashion. Lévy walks result in enhanced diffusion, ie, diffusion that grows as t α, α> 1. When applied to the description of a passive scalar diffusing in a fluctuating fluid flow the model generalizes Taylor’s correlated-walk approach. It yields Richardson’s t 3 law for the turbulent diffusion of a passive scalar in a Kolmogorov-(5/3) homogeneous turbulent flow and also gives the deviations from the (5/3) exponent resulting from Mandelbrot’s intermittency. The model can be extended to studies of chemical reactions in turbulent flow.