Recent advancements in the electrospinning method enable the production of ultrafine solid and continuous fibers with diameters ranging from a few nanometers to a few hundred nanometers with controlled surface and morphological features. A wide range of biopolymers can be electrospun into mats with a specific fiber arrangement and structural integrity. These features of nanofiber mats are morphologically similar to the extracellular matrix of natural tissues, which are characterized by a wide pore diameter distribution, a high porosity, effective mechanical properties, and specific biochemical properties. This has resulted in various kinds of applications for polymer nanofibers in the field of biomedicine and biotechnology. The current emphasis of research is on exploiting these properties and focusing on determining the appropriate conditions for electrospinning various biopolymers for biomedical applications, including scaffolds used in tissue engineering, wound dressing, drug delivery, artificial organs, and vascular grafts, and for protective shields in specialty fabrics. This paper reviews the research on biomedical applications of electrospun nanofibers.