Polymeric nanofibers have appeared as unique one-dimensional nanomaterials. Nanofibers possess exceptionally high specific surface area and essential properties. Various polymeric nanofibers and nanocomposite nanofibers have been developed using thermoplastic and thermosetting matrices and organic and inorganic nanoparticles. This review documents the worth of nanofiber technology regarding the synthesis, morphology, physical properties, and applications of the nanostructures. The nanofiber surface morphology and diameter directly influence the physical characteristics such as electrical conductivity, mechanical properties, and thermal stability. Fiber processing techniques and related parameters also affect their texture and properties. In this regard, electrospinning is frequently used to form polymeric nanofibers and nanocomposite nanofibers. Electrospun nanofibers having large surface area and excellent fiber orientation, alignment, and morphology. The polymer nanocomposites with different nanofillers (carbon nanotube, graphene, carbon nanofibers, and other nanoparticles) have been processed into high performance electrospun nanofibers. Electrospun nanomaterials have been applied in engineered structures, nanofibrous membranes, electronic devices, tissue engineering, drug delivery, antibacterial materials and other biomedical applications. Henceforth, this is a comprehensive review on physical characteristics, morphology, processing aspects, and application areas for polymeric and nanocomposite nanofibers.