  Gauge theories represent a fundamental framework underlying modern physics,
constituting the basis of the Standard Model and also providing useful
descriptions of various phenomena in condensed matter. Realizing gauge theories
on accessible and tunable tabletop quantum devices offers the possibility to
study their dynamics from first principles time evolution and to probe their
exotic physics, including that generated by deviations from gauge invariance,
which is not possible, e.g., in dedicated particle colliders. Not only do
cold-atom quantum simulators hold the potential to provide new insights into
outstanding high-energy and nuclear-physics questions, they also provide a
versatile tool for the exploration of topological phases and
ergodicity-breaking mechanisms relevant to low-energy many-body physics. In
recent years, cold-atom quantum simulators have demonstrated impressive
progress in the large-scale implementation of $1+1$D Abelian gauge theories. In
this Review, we chronicle the progress of cold-atom quantum simulators of gauge
theories, highlighting the crucial advancements achieved along the way in order
to reliably stabilize gauge invariance and go from building blocks to
large-scale realizations where \textit{bona fide} gauge-theory phenomena can be
probed. We also provide a brief outlook on where this field is heading, and
what is required experimentally and theoretically to bring the technology to
the next level by surveying various concrete proposals for advancing these
setups to higher spatial dimensions, higher-spin representations of the gauge
field, and non-Abelian gauge groups.
