  The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is a novel superconducting
state in a strong magnetic field characterized by the formation of Cooper pairs
with nonzero total momentum (k \uparrow, -k+q \downarrow), instead of the
ordinary BCS pairs (k \uparrow, -k \downarrow). A fascinating aspect of the
FFLO state is that it exhibits inhomogeneous superconducting phases with a
spatially oscillating order parameter and spin polarization. The FFLO state has
been of interest in various research fields, not only in superconductors in
solid state physics, but also in neutral Fermion superfluid of ultracold atomic
gases and in color superconductivity in high energy physics. In spite of
extensive studies of various superconductors, there has been no undisputed
experimental verification of the FFLO state, mainly because of the very
stringent conditions required of the superconducting materials. Among several
classes of materials, certain heavy fermion and organic superconductors are
believed to provide conditions that are favorable to the formation of the FFLO
state. This review presents recent experimental and theoretical developments of
the FFLO state mainly in heavy fermion superconductors. In particular we
address the recently discovered quasi-two-dimensional superconductor CeCoIn_5,
which is a strong candidate for the formation of the FFLO state.
