  The dichotomy between fermions and bosons is at the root of many physical
phenomena, from metallic conduction of electricity to super-fluidity, and from
the periodic table to coherent propagation of light. The dichotomy originates
from the symmetry of the quantum mechanical wave function to the interchange of
two identical particles. In systems that are confined to two spatial dimensions
particles that are neither fermions nor bosons, coined "anyons", may exist. The
fractional quantum Hall effect offers an experimental system where this
possibility is realized. In this paper we present the concept of anyons, we
explain why the observation of the fractional quantum Hall effect almost forces
the notion of anyons upon us, and we review several possible ways for a direct
observation of the physics of anyons. Furthermore, we devote a large part of
the paper to non-abelian anyons, motivating their existence from the point of
view of trial wave functions, giving a simple exposition of their relation to
conformal field theories, and reviewing several proposals for their direct
observation.
