In the fields of light manipulation and localization, quasiperiodic photonic crystals, or photonic quasicrystals (PQs), are causing an upsurge in research because of their rotational symmetry and long-range orientation of transverse lattice arrays, as they lack translational symmetry. It allows for the optimization of well-established light propagation properties and has introduced new guiding features. Therefore, as a class, quasiperiodic photonic crystal fibers, or photonic quasicrystal fibers (PQFs), are considered to add flexibility and richness to the optical properties of fibers and are expected to offer significant potential applications to optical fiber fields. In this review, the fundamental concept, working mechanisms, and invention history of PQFs are explained. Recent progress in optical property improvement and its novel applications in fields such as dispersion control, polarization-maintenance, supercontinuum generation, orbital angular momentum transmission, plasmon-based sensors and filters, and high nonlinearity and topological mode transmission, are then reviewed in detail. Bandgap-type air-guiding PQFs supporting low attenuation propagation and regulation of photonic density states of quasiperiodic cladding and in which light guidance is achieved by coherent Bragg scattering are also summarized. Finally, current challenges encountered in the guiding mechanisms and practical preparation techniques, as well as the prospects and research trends of PQFs, are also presented.