We review various theoretical approaches to the states of quantum chromodynamics (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 the non-perturbative methods; the strong-coupling expansion, the large-N_c 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 bears 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.
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