  In this review, we summarize recent progresses on the possible phases of
quantum chromodynamics (QCD) in the presence of a strong magnetic field, mainly
from the views of chiral effective Nambu--Jona-Lasinio model. Four kinds of
phase transitions are explored in detail: chiral symmetry breaking and
restoration, neutral pseudoscalar superfluidity, charged pion superfluidity and
charged rho superconductivity. In particular, we revisit the unsolved problems
of inverse magnetic catalysis effect and the competition between chiral density
wave and solitonic modulation, and eventually manage to obtain useful results
by adopting self-consistent schemes.
