  The quantum-mechanical description of assemblies of particles whose motion is
confined to two (or one) spatial dimensions offers many possibilities that are
distinct from bosons and fermions. We call such particles anyons. The simplest
anyons are parameterized by an angular phase parameter $\theta$. $\theta = 0,
\pi$ correspond to bosons and fermions respectively; at intermediate values we
say that we have fractional statistics. In two dimensions, $\theta$ describes
the phase acquired by the wave function as two anyons wind around one another
counterclockwise. It generates a shift in the allowed values for the relative
angular momentum. Composites of localized electric charge and magnetic flux
associated with an abelian U(1) gauge group realize this behavior. More complex
charge-flux constructions can involve non-abelian and product groups acting on
a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual
statistics. Interchanges of non-abelian anyons implement unitary
transformations of the wave function within an emergent space of internal
states. Anyons of all kinds are described by quantum field theories that
include Chern--Simons terms. The crossings of one-dimensional anyons on a ring
are uni-directional, such that a fractional phase $\theta$ acquired upon
interchange gives rise to fractional shifts in the relative momenta between the
anyons. The quasiparticle excitations of fractional quantum Hall states have
long been predicted to include anyons. Recently the anyon behavior predicted
for quasiparticles in the $\nu = 1/3$ fractional quantum Hall state has been
observed both in scattering and in interferometric experiments. Excitations
within designed systems, notably including superconducting circuits, can
exhibit anyon behavior. Such systems are being developed for possible use in
quantum information processing.
