  This article reviews theoretical and experimental developments for
one-dimensional Fermi gases. Specifically, the experimentally realized
two-component delta-function interacting Fermi gas -- the Gaudin-Yang model --
and its generalisations to multi-component Fermi systems with larger spin
symmetries. The exact results obtained for Bethe ansatz integrable models of
this kind enable the study of the nature and microscopic origin of a wide range
of quantum many-body phenomena driven by spin population imbalance, dynamical
interactions and magnetic fields. This physics includes
Bardeen-Cooper-Schrieffer-like pairing, Tomonaga-Luttinger liquids, spin-charge
separation, Fulde-Ferrel-Larkin-Ovchinnikov-like pair correlations, quantum
criticality and scaling, polarons and the few-body physics of the trimer state
(trions). The fascinating interplay between exactly solved models and
experimental developments in one dimension promises to yield further insight
into the exciting and fundamental physics of interacting Fermi systems.
