Chaotic transport is studied experimentally in a two-dimensional flow in a rapidly rotating annular tank. The flow consists of a chain of vortices sandwiched between two azimuthal jets. Automated image processing techniques are used to track the motions of neutrally buoyant tracer particles suspended in the flow. If the flow has periodic time dependece, the tracers typically follow chaotic trajectories, alternately sticking in vortices and flying for long distances in the jets. Probability distribution functions (PDFs) are measured for sticking and flight times. The flight PDFs are power laws, indicated in some cases that the particle motion can be characterized as Levy flights (with a divergent second moment for flight times). The variance of an ensemble of particles is found to vary in time as sigma^2 ~ t^d, with d > 1 (superdiffusion). The dependence of the variance exponent d on the flight and sticking PDFs is studied and found to be consistent with calculations relating Levy flights and anomalous diffusion (d not equal 1). In a turbulent flow, Levy flights no longer are present and the mixing appears to be normally diffusive (d=1). A review of previous experiments on anomalous diffusion is included.