  Recent development of inhomogeneous chiral phase in Kyoto group is briefly
reviewed. First, the nesting effect of the Fermi surface is emphasized as a key
mechanism leading to inhomogeneous chiral phase. After introduction of
inhomogeneous chiral phase, some topological aspect is discussed in 1+1
dimensions and in the presence of the magnetic field in 1+3 dimensions.
Spectral asymmetry gives rise to anomalous quark number and is closely related
to chiral anomaly. It may induce spontaneous magnetization and a novel Lifshitz
point may appear on the line $\mu=0$. Some astronomical implications are
briefly discussed. Finally, the effects of the fluctuations around the order
parameter are discussed; the propagator of the fluctuations exhibits a
singularity at finite momentum and changes the properties of the phase
transition through loop diagrams.
