  The scattering matrix (S-matrix), relating the initial and final states of a
physical system undergoing a scattering process, is a fundamental object in
quantum mechanics and quantum field theory. The study of factorised S-matrices,
in which many-body scattering factorizes into a product of two-body terms to
yield an integrable model, has long been considered the domain of mathematical
physics. Many beautiful results have been obtained over several decades for
integrable models of this kind, with far reaching implications in both
mathematics and theoretical physics. The viewpoint that these were only toy
models changed dramatically with brilliant experimental advances in realizing
low-dimensional quantum many-body systems in the lab. These recent experiments
involve both the traditional setting of condensed matter physics and the
trapping and cooling of atoms in optical lattices to engineer and study
quasi-one-dimensional systems. In some cases the quantum physics of
one-dimensional systems is arguably more interesting than their
three-dimensional counterparts, because the effect of interactions is more
pronounced when atoms are confined to one dimension. This article provides a
brief overview of these ongoing developments, which highlight the fundamental
importance of integrability.
