  The magnetohydrodynamic (MHD) description of plasmas with relativistic
particles necessarily includes an additional new field, the chiral chemical
potential associated with the axial charge (i.e., the number difference between
right- and left-handed relativistic fermions). This chiral chemical potential
gives rise to a contribution to the electric current density of the plasma
(\emph{chiral magnetic effect}). We present a self-consistent treatment of the
\emph{chiral MHD equations}, which include the back-reaction of the magnetic
field on a chiral chemical potential and its interaction with the plasma
velocity field. A number of novel phenomena are exhibited. First, we show that
the chiral magnetic effect decreases the frequency of the Alfv\'{e}n wave for
incompressible flows, increases the frequencies of the Alfv\'{e}n wave and of
the fast magnetosonic wave for compressible flows, and decreases the frequency
of the slow magnetosonic wave. Second, we show that, in addition to the
well-known laminar chiral dynamo effect, which is not related to fluid motions,
there is a dynamo caused by the joint action of velocity shear and chiral
magnetic effect. In the presence of turbulence with vanishing mean kinetic
helicity, the derived mean-field chiral MHD equations describe turbulent
large-scale dynamos caused by the chiral alpha effect, which is dominant for
large fluid and magnetic Reynolds numbers. The chiral alpha effect is due to an
interaction of the chiral magnetic effect and fluctuations of the small-scale
current produced by tangling magnetic fluctuations (which are generated by
tangling of the large-scale magnetic field by sheared velocity fluctuations).
These dynamo effects may have interesting consequences in the dynamics of the
early universe, neutron stars, and the quark--gluon plasma.
