Nanoclays are layered mineral silicates, i.e., layered silicate nanosheets. Nanoclays such as montmorillonite, bentonite, kaolinite, etc., have been used as reinforcements in the nanofibers. Numerous polymers have been used to fabricate the nanofibers, including poly(vinylidene fluoride), poly(vinyl alcohol), polycaprolactone, nylon, polyurethane, poly(ethylene oxide), and others. To develop better compatibility with polymers, nanoclays have been organo-modified prior to reinforcement in the nanofiber matrices. This state-of-the-art review highlights the fundamentals, design, fabrication, and characteristics of the polymer/nanoclay nanofibers. The nanoclay filled nanocomposite nanofibers have been fabricated using electrospinning and other fiber processing techniques. The electrospinning technique has been preferred to form the nanoclay-filled nanofibers, owing to the better control of processing parameters and resulting nanofiber properties. The electrospun polymer/nanoclay nanofibers usually have fine nanoparticle dispersions, microstructures, smooth textures, and narrow diameters. The physical properties of the designed nanofibers depend upon the processing technology used, solvent, solution/melt concentration, flow rate, spinning speed, voltage, and other process parameters. Hence, this review attempts to assess a literature-driven consequence of embedding nanoclays in the polymeric nanofibers in a broad context of the application of these fibrous materials. Conclusively, to design the polymer/nanoclay nanofibers, montmorillonite nanoclay has been observed as a nanofiller in most of the studies, and, similarly, the electrospinning technique was preferred as a fabrication technique. Almost all the physical properties of the nanofibers studied revealed dependences upon the choice of the polymer matrix for nanofiber formation as well as the nanoclay contents, modification, and dispersion state. Accordingly, the nylon/nanoclay nanofibers have been investigated for nanofiller dispersion, mechanical properties, and thermal profiles. The antibacterial properties were among the prominent features of the poly(vinyl alcohol)/nanoclay nanofibers. The poly(vinylidene fluoride)/nanoclay systems were explored for the microstructure, crystallinity, and piezoelectric properties. The polycaprolactone/nanoclay nanofibers having fine microstructure were capable of forming tissue engineering scaffolds. The drug delivery and sound absorption properties were noticeable for the polyurethane/nanoclay nanofiber systems. Moreover, the poly(lactic acid)/nanoclay nanofibers were found to have prominent biodegradability and low gas permeability features. The resulting polymer/nanoclay nanocomposite nanofiber systems found potential for the technical applications of sensors, packaging, tissue engineering, and wound healing. However, thorough research efforts have been found to be desirable to find the worth of polymer/nanoclay nanofibers in several concealed technological sectors of energy, electronics, aerospace, automotives, and biomedical fields.