  In recent years, studies of long-range interacting (LRI) systems have taken
centre stage in the arena of statistical mechanics and dynamical system
studies, due to new theoretical developments involving tools from as diverse a
field as kinetic theory, non-equilibrium statistical mechanics, and large
deviation theory, but also due to new and exciting experimental realizations of
LRI systems. In this invited contribution, we discuss the general features of
long-range interactions, emphasizing in particular the main physical phenomenon
of non-additivity, which leads to a plethora of distinct effects, both
thermodynamic and dynamic, that are not observed with short-range interactions:
Ensemble inequivalence, slow relaxation, broken ergodicity. We also discuss
several physical systems with long-range interactions: mean-field spin systems,
self-gravitating systems, Euler equations in two dimensions, Coulomb systems,
one-component electron plasma, dipolar systems, free-electron lasers, atoms
trapped in optical cavities.
