  A range of quantum field theoretical phenomena driven by external magnetic
fields and their applications in relativistic systems and quasirelativistic
condensed matter ones, such as graphene and Dirac/Weyl semimetals, are
reviewed. We start by introducing the underlying physics of the magnetic
catalysis. The dimensional reduction of the low-energy dynamics of relativistic
fermions in an external magnetic field is explained and its role in catalyzing
spontaneous symmetry breaking is emphasized. The general theoretical
consideration is supplemented by the analysis of the magnetic catalysis in
quantum electrodynamics, chromodynamics and quasirelativistic models relevant
for condensed matter physics. By generalizing the ideas of the magnetic
catalysis to the case of nonzero density and temperature, we argue that other
interesting phenomena take place. The chiral magnetic and chiral separation
effects are perhaps the most interesting among them. In addition to the general
discussion of the physics underlying chiral magnetic and separation effects, we
also review their possible phenomenological implications in heavy-ion
collisions and compact stars. We also discuss the application of the magnetic
catalysis ideas for the description of the quantum Hall effect in monolayer and
bilayer graphene, and conclude that the generalized magnetic catalysis,
including both the magnetic catalysis condensates and the quantum Hall
ferromagnetic ones, lies at the basis of this phenomenon. We also consider how
an external magnetic field affects the underlying physics in a class of
three-dimensional quasirelativistic condensed matter systems, Dirac semimetals.
While at sufficiently low temperatures and zero density of charge carriers,
such semimetals are expected to reveal the regime of the magnetic catalysis,
the regime of Weyl semimetals with chiral asymmetry is realized at nonzero
density...
