  We discuss the possible existence of a superconducting phase at high magnetic
field in organic quasi-one-dimensional conductors. We consider in particular
(i) the formation of a Larkin-Ovchinnikov-Fulde-Ferrell state, (ii) the role of
a temperature-induced dimensional crossover occuring when the transverse
coherence length $\xi_z(T)$ becomes of the order of the lattice spacing, and
(iii) the effect of a magnetic-field-induced dimensional crossover resulting
from the localization of the wave functions at high magnetic field. In the case
of singlet spin pairing, only the combination of (i) and (iii) yields a picture
consistent with recent experiments in the Bechgaard salts showing the existence
of a high-field superconducting phase. We point out that the vortex lattice is
expected to exhibit unusual characteristics at high magnetic field.
