  Diffusion occurs in numerous physical systems throughout nature, drawing its
generality from the universality of the central limit theorem. Around a century
ago it was realized that an extension to this type of dynamics can be obtained
in the form of ``anomalous" diffusion where distributions are allowed to have
heavy, power-law tails. Due to a unique feature of its momentum-dependant
dissipative friction force, laser-cooled atomic ensembles can be used as a test
bed for such dynamics. The interplay between laser cooling and anomalous
dynamics bears deep, predictive implications for fundamental concepts in both
equilibrium and non-equilibrium statistical physics. The high degree of control
available in cold-atom experiments allows for tuning of the parameters of the
friction force, revealing transitions in the dynamical properties of the
system. Rare events, in both the momentum and spatial distributions, are
described by non-normalized states using tools adapted from infinite ergodic
theory. This leads to new experimental and theoretical results, illuminating
the various features of the system.
