We use a rapidly rotating tank filled with water to study random walks of tracer particles in two different flows: one with coherent structures (vortices and jets), and one without (weakly turbulent). Most random walks, such as those taken by dye molecules diffusing by Brownian motion, obey the Central Limit Theorem; their motion can be characterized by a diffusion constant. The random walks in our experiment with coherent structures are Levy flights: random walks with infinite mean square step size. Levy flights are particularly interesting as the Central Limit Theorem does not apply to them; the effective diffusion constant becomes infinite. Tracer particles in the weakly turbulent flow diffuse normally. We discuss our observations and mathematical descriptions of transport in situations with Levy flights.