  Relativistic magnetohydrodynamics (RMHD) provides an extremely useful
description of the low-energy long-wavelength phenomena in a variety of
physical systems from quark-gluon plasma in heavy-ion collisions to matters in
supernovas, compact stars, and early universe. We review the recent theoretical
progresses of RMHD, such as a formulation of RMHD from the perspective of
magnetic flux conservation using the entropy-current analysis, the
nonequilibrium statistical operator approach applied to quantum
electrodynamics, and the relativistic kinetic theory. We discuss how the
transport coefficients in RMHD are computed in kinetic theory and perturbative
quantum field theories. We also explore the collective modes and instabilities
in RMHD with a special emphasis on the role of chirality in a parity-odd
plasma. We also give some future prospects of RMHD, including the interaction
with spin hydrodynamics and the new kinetic framework with magnetic flux
conservation.
