  We study the phase diagram of the one dimensional (1D) $U-V$ model at quarter
filling in the most general case where the on-site and first-neighbour
interactions $U$ and $V$ can be both attractive and repulsive. The results have
been obtained using exact diagonalization of small clusters and variational
techniques, as well as exact results in various limits. We have analyzed four
properties of the groundstate: i)~whether it is insulating or metallic;
\hbox{ii)~whether} it is homogenous or phase separated; iii)~whether it has a
spin gap; iv)~whether it has dominant superconducting fluctuations. With eight
phases, the resulting phase diagram is unexpectedly rich. The four phases not
found in the weak coupling limit are: i) an insulating phase when $U$ and $V$
are large enough; ii) a region of phase separation when $V$ is attractive; iii)
another region of phase separation when $V$ is large enough and $U$ small; iv)
a region with dominant superconducting fluctuations when $V$ is intermediate
and $U$ small. The actual nature of this last phase, which has pairs but no
spin gap, is not fully clear yet.
