  We review various theoretical approaches to the states of QCD matter out of
quarks and gluons in extreme environments such as the high-temperature states
at zero and finite baryon density and the dimensionally reduced state under an
intense magnetic field. The topics at high temperature include the Polyakov
loop and the 't Hooft loop in the perturbative regime, the Polyakov loop
behaviour and the phase transition in some of non-perturbative methods; the
strong-coupling expansion, the large-Nc limit and the holographic QCD models.
These analyses are extended to hot and dense matter with a finite baryon
chemical potential. We point out that the difficulty in the finite-density
problem has similarity to that under a strong magnetic field. We make a brief
summary of results related to the topological contents probed by the magnetic
field and the Chiral Magnetic Effect. We also address the close connection to
the (1+1) dimensional system.
