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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, Möbius 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 Möbius insulator with massless surface modes that resemble the topological structure of a Möbius twist (dubbed as Möbius fermions); (4) a time-reversal-invariant version of Möbius 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.