  We review the concepts and the present state of theoretical studies of
spin-imbalanced superfluidity, in particular the elusive
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold
quantum gases. The comprehensive presentation of the theoretical basis for the
FFLO state that we provide is useful also for research on the interplay between
magnetism and superconductivity in other physical systems. We focus on settings
that have been predicted to be favourable for the FFLO state, such as optical
lattices in various dimensions and spin-orbit coupled systems. These are also
the most likely systems for near-future experimental observation of the FFLO
state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and
experimentally important limitations, such as finite-size effects and trapping
potentials, are considered. In addition, we provide a comprehensive review of
the various ideas presented for the observation of the FFLO state. We conclude
our review with an analysis of the open questions related to the FFLO state,
such as its stability, superfluid density, collective modes and extending the
FFLO superfluid concept to new types of lattice systems.
