Several experiments are being conducted at heavy-ion colliders around the world to determine the location of the proposed critical end point of quantum chromodynamics (QCD) in the T - μB phase diagram. As the presence of a very strong magnetic field is relevant to peripheral heavy-ion collisions, magnetars, and the early Universe, it is important to investigate the effect of a high magnetic field strength on QCD phase diagrams. We summarize the recent status and new developments in studies investigating QCD phase transitions under an extremely strong magnetic field. By doing so, we believe that this work will promote both theoretical and experimental research in this field. The T - B phase diagrams are produced by Lattice QCD simulations. Other phase diagrams (E - B, μB - B, μI - B, and Ω - B) are mainly studied by using the chiral effective Nambu Jona-Lasinio model. A rotating magnetic field is adopted for the study of color superconductivity. The Ginzburg-Landau approximation is used to study π-superfluidity and ρ-superconductivity in a very strong magnetic field. Physical effects, besides a magnetic field B, can also be measured when sketching a QCD phase diagram, such as temperature T, strong electric field E, chemical potentials μ, and rotational angular velocity Ω. We present five QCD phase diagrams: T - B, E - B, μB - B, μI - B, and Ω - B. The following phases are present in many (if not all) of the five QCD phase diagrams: chiral symmetry breaking, chiral symmetry restoration, inhomogeneous chiral phase, π0-condensation, π-superfluidity, ρ-superconductivity, and color superconductivity. The running of the coupling constant with magnetic field is consistent with the decrease of the pseudo-critical deconfinement temperature, providing a natural explanation for the inverse magnetic catalysis effect. We also found that a chiral anomaly induces pseudoscalar condensation in a parallel electromagnetic field, and that there appears to be a chiral-symmetry restoration phase in the E - B phase diagram. Without consideration of confinement, color superconductivity is typically favored for large baryon chemical potential; however, chiral density wave is also possible in the large B and relatively small μB region of the phase diagram. In an external magnetic field, the π-superfluid with finite isospin chemical potential acts similarly to a Type-II superconductor with finite electric chemical potential. Both π-superfluidity and ρ-superconductivity are possible in a parallel magnetic field and rotation, but the latter is more favored for larger Ω particles. © 2023 Science Press. All rights reserved.
