  The properties of a quasi-one-dimensional (quasi-1D) superconductor with {\it
an open Fermi surface} are expected to be unusual in a magnetic field. On the
one hand, the quasi-1D structure of the Fermi surface strongly favors the
formation of a non-uniform state (Larkin-Ovchinnikov-Fulde-Ferrell (LOFF)
state) in the presence of a magnetic field acting on the electron spins. On the
other hand, a magnetic field acting on an open Fermi surface induces a
dimensional crossover by confining the electronic wave-functions wave-functions
along the chains of highest conductivity, which results in a divergence of the
orbital critical field and in a stabilization at low temperature of a cascade
of superconducting phases separated by first order transistions. In this paper,
we study the phase diagram as a function of the anisotropy. We discuss in
details the experimental situation in the quasi-1D organic conductors of the
Bechgaard salts family and argue that they appear as good candidates for the
observation of the LOFF state, provided that their anisotropy is large enough.
Recent experiments on the organic quasi-1D superconductor (TMTSF)$_2$ClO$_4$
are in agreement with the results obtained in this paper and could be
interpreted as a signature of a high-field superconducting phase. We also point
out the possibility to observe a LOFF state in some quasi-2D organic
superconductors.
