  In this review we discuss a wide range of topological properties of electron
quasiparticles in Dirac and Weyl semimetals. Their nontrivial topology is
quantified by a monopole-like Berry curvature in the vicinity of Weyl nodes, as
well as by the energy and momentum space separations between the nodes. The
momentum separation, which is also known as the chiral shift, is one of the key
elements of this review. We show that it can be dynamically generated in Dirac
materials in a background magnetic field. We also pay a special attention to
various forms of interplay between the background electromagnetic fields and
the topological characteristics of Dirac and Weyl semimetals. In particular, we
discuss their signature features in the transport of the electric and chiral
charges, heat, as well as the quantum oscillations associated with the Fermi
arc states. The origin of the dissipative transport of the Fermi arc states is
critically examined. Finally, a consistent chiral kinetic theory for the
description of Weyl semimetals is reviewed and its applications are
demonstrated.
