  We present a summary of the recent advances achieved in the study of three
flavor Larkin-Ovchinnikov-Fulde-Ferrell (LOFF)phase of QCD. We have explored,
using a Ginzburg-Landau expansion of the free energy, the LOFF phase with three
flavors, in the simplest single plane wave structure, using the NJL
four-fermion coupling. We have found that this phase does not suffer the
chromo-magnetic instability problem. A preliminary study of astrophysical
effects of quark matter in the aforementioned phase has been done and we have
evaluated self-consistently the strange quark mass extending the pairing ansatz
to the CubeX and 2Cube45z structure. Finally we have investigated the
possibility of Goldstone bosons condensation in the favored cubic structures of
LOFF phase.
