  Electrons in low-temperature solids are governed by the non-relativistic
Schr$\ddot{o}$dinger equation, since the electron velocities are much slower
than the speed of light. Remarkably, the low-energy quasi-particles given by
electrons in various materials can behave as relativistic Dirac/Weyl fermions
that obey the relativistic Dirac/Weyl equation. We refer to these materials as
"Dirac/Weyl materials", which provide a tunable platform to test relativistic
quantum phenomena in table-top experiments. More interestingly, different types
of physical fields in these Weyl/Dirac materials, such as magnetic
fluctuations, lattice vibration, strain, and material inhomogeneity, can couple
to the "relativistic" quasi-particles in a similar way as the $U(1)$ gauge
coupling. As these fields do not have gauge-invariant dynamics in general, we
refer to them as "pseudo-gauge fields". In this chapter, we overview the
concept and physical consequences of pseudo-gauge fields in Weyl/Dirac
materials. In particular, we will demonstrate that pseudo-gauge fields can
provide a unified understanding of a variety of physical phenomena, including
chiral zero modes inside a magnetic vortex core of magnetic Weyl semimetals, a
giant current response at magnetic resonance in magnetic topological
insulators, and piezo-electromagnetic response in time-reversal invariant
systems. It turns out that these phenomena are deeply related to various
concepts in high-energy physics, such as chiral anomaly and axion
electrodynamics.
