  The process of diffusion is the most elementary stochastic transport process.
Brownian motion, the representative model of diffusion, played a important role
in the advancement of scientific fields such as physics, chemistry, biology and
finance. However, in recent decades, non-diffusive transport processes with
non-Brownian statistics were observed experimentally in a multitude of
scientific fields. Examples include human travel, in-cell dynamics, the motion
of bright points on the solar surface, the transport of charge carriers in
amorphous semiconductors, the propagation of contaminants in groundwater, the
search patterns of foraging animals and the transport of energetic particles in
turbulent plasmas. These examples showed that the assumptions of the classical
diffusion paradigm, assuming an underlying uncorrelated (Markovian), Gaussian
stochastic process, need to be relaxed to describe transport processes
exhibiting a non-local character and exhibiting long-range correlations.
  This article does not aim at presenting a complete review of non-diffusive
transport, but rather an introduction for readers not familiar with the topic.
For more in depth reviews, we recommend some references in the following.
First, we recall the basics of the classical diffusion model and then we
present two approaches of possible generalizations of this model: the
Continuous-Time-Random-Walk (CTRW) and the fractional L\'evy motion (fLm).
