Single-crystal fiber (SCF) is a fiber-shaped monocrystalline material, which maintains the outstanding physical and chemical properties of bulk crystals and simultaneously exhibits the excellent thermal dissipation efficiency and waveguide structure of optical fibers, therefore indicating significant potential applications in solid-state lasers, scintillation detectors, high temperature sensors, etc. Recently, numerous researches have been focused on the crystal growth, the fabrication of cladding structure as well as applications of SCFs. In this paper, the characteristics and developments of the most widely applied crystal growth techniques of SCFs, including micro-pulling-down method, laser-heated pedestal growth technique and edge-defined film-fed growth technique, are introduced in detail. Recent progress of several research groups engaging in SCFs both at home and abroad are also discussed. The investigation of applications of SCFs in high-power laser systems, mid-infrared lasers as well as scintillation detectors are analyzed. The strategies of cladding fabrications is also introduced briefly.