  Quantum materials hosting Weyl fermions have opened a new era of research in
condensed matter physics. First proposed in 1929 in particle physics, Weyl
fermions have yet to be observed as elementary particles. In 2015, Weyl
fermions were detected as collective electronic excitations in the strong
spin-orbit coupled material tantalum arsenide, TaAs. This discovery was
followed by a flurry of experimental and theoretical explorations of Weyl
phenomena in materials. Weyl materials naturally lend themselves to the
exploration of the topological index associated with Weyl fermions and their
divergent Berry curvature field, as well as the topological bulk-boundary
correspondence giving rise to protected conducting surface states. Here, we
review the broader class of Weyl topological phenomena in materials, starting
with the observation of emergent Weyl fermions in the bulk and of Fermi arc
states on the surface of the TaAs family of crystals by photoemission
spectroscopy. We then discuss some of the exotic optical and magnetic responses
observed in these materials, as well as the progress in developing some of the
related chiral materials. We discuss the conceptual development of high-fold
chiral fermions, which generalize Weyl fermions, and we review the observation
of high-fold chiral fermion phases by taking the rhodium silicide, RhSi, family
of crystals as a prime example. Lastly, we discuss recent advances in Weyl-line
phases in magnetic topological materials. With this Review, we aim to provide
an introduction to the basic concepts underlying Weyl physics in condensed
matter, and to representative materials and their electronic structures and
topology as revealed by spectroscopic studies. We hope this work serves as a
guide for future theoretical and experimental explorations of chiral fermions
and related topological quantum systems with potentially enhanced
functionalities.
