  In this review, we summarize recent progress on the possible phases of
quantum chromodynamics (QCD) in the presence of a strong magnetic field, mainly
from the views of the 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 competition between the chiral density
wave and solitonic modulation phases. It is shown that useful results can be
obtained by adopting self-consistent schemes.
