Advection of tracers is studied numerically in time-dependent, two-dimensional cellular flows and a time-independent, three-dimensional cellular flow field. Tracers in these flows follow trajectories that are either periodic or chaotic and mimic correlated Lévy flights. The probability density function of displacements for particles in the ordered regions of the flow follows a classical Gaussian dispersion process. The particle trajectories in the chaotic regions of the flow exhibit anomalous diffusion and the probability density function of displacements is well modeled by a time-fractional diffusion equation of order alpha. The overall process of particle dispersion is found to be controlled mainly by the chaotic regions within the flow field. From the perspective of Lagrangian dynamics our results indicate that the advection of particles in flow fields prone to exhibit chaotic advection is a combination of both classical, i.e., Gaussian, behavior and anomalous, i.e., non-Gaussian, diffusion.