  The interplay of quantum anomalies with magnetic field and vorticity results
in a variety of novel non-dissipative transport phenomena in systems with
chiral fermions, including the quark-gluon plasma. Among them is the Chiral
Magnetic Effect (CME) -- the generation of electric current along an external
magnetic field induced by chirality imbalance. Because the chirality imbalance
is related to the global topology of gauge fields, the CME current is
topologically protected and hence non-dissipative even in the presence of
strong interactions. As a result, the CME and related quantum phenomena affect
the hydrodynamical and transport behavior of strongly coupled quark-gluon
plasma, and can be studied in relativistic heavy ion collisions where strong
magnetic fields are created by the colliding ions. Evidence for the CME and
related phenomena has been reported by the STAR Collaboration at Relativistic
Heavy Ion Collider at BNL, and by the ALICE Collaboration at the Large Hadron
Collider at CERN. The goal of the present review is to provide an elementary
introduction into the physics of anomalous chiral effects, to describe the
current status of experimental studies in heavy ion physics, and to outline the
future work, both in experiment and theory, needed to eliminate the existing
uncertainties in the interpretation of the data.
