  A lattice symmetry, if being nonsymmorphic, is defined by combining a point
group symmetry with a fractional lattice translation that cannot be removed by
changing the lattice origin. Nonsymmorphic symmetry has a substantial influence
on both the connectivity and topological properties of electronic band
structures in solid-state quantum materials. In this article, we review how
nonsymmorphic crystalline symmetries can drive and further protect the
emergence of exotic fermionic quasiparticles, including Dirac, M\"obius and
hourglass fermions, that manifest as the defining energy band signatures for a
plethora of gapless or gapped topological phases of matter. We first provide a
classification of energy band crossings in crystalline solids, with an emphasis
on symmetry-enforced band crossings that feature a nonsymmorphic-symmetry
origin. In particular, we will discuss four distinct classes of
nonsymmorphic-symmetry-protected topological states as well as their signature
fermionic modes: (1) a $Z_2$ topological nonsymmorphic crystalline insulator
with two-dimensional surface Dirac fermions; (2) a Dirac semimetal with three
dimensional bulk-state Dirac nodes pinned at certain high symmetry momenta; (3)
a topological M\"obius insulator with massless surface modes that resemble the
topological structure of a M\"obius twist (dubbed as M\"obius fermions); (4) a
time-reversal-invariant version of M\"obius insulators with hourglass-shaped
massless surface states (dubbed as hourglass fermions). The emergence of the
above exotic topological matter perfectly demonstrates the crucial roles of
both symmetry and topology in understanding the colorful world of quantum
materials.

Mesoscale and Nanoscale Physics;Materials Science;Strongly Correlated Electrons