  A review of non-diffusive transport in fluids and plasmas is presented. In
the fluid context, non-diffusive chaotic transport by Rossby waves in zonal
flows is studied following a Lagrangian approach. In the plasma physics context
the problem of interest is test particle transport in pressure-gradient-driven
plasma turbulence. In both systems the probability density function (PDF) of
particle displacements is strongly non-Gaussian and the statistical moments
exhibit super-diffusive anomalous scaling. Fractional diffusion models are
proposed and tested in the quantitative description of the non-diffusive
Lagrangian statistics of the fluid and plasma problems. Also, fractional
diffusion operators are used to construct non-local transport models exhibiting
up-hill transport, multivalued flux-gradient relations, fast pulse propagation
phenomena, and "tunneling" of perturbations across transport barriers.
