Wounds are a consequence of disruption in the structure, integrity, or function of the skin or tissue. Once a wound is formed following mechanical or chemical damage, the process of wound healing is initiated, which involves a series of chemical signaling and cellular mechanisms that lead to regeneration and/or repair. Disruption in the healing process may result in complications; therefore, interventions to accelerate wound healing are essential. In addition to mechanical support provided by sutures and traditional wound dressings, therapeutic agents play a major role in accelerating wound healing. The medicines known to improve the rate and extent of wound healing include antibacterial, anti-inflammatory, and proliferation enhancing agents. Nonetheless, the development of these agents into eluting nanofibers presents the possibility of fabricating wound dressings and sutures that provide mechanical support with the added advantage of local delivery of therapeutic agents to the site of injury. Herein, the process of wound healing, complications of wound healing, and current practices in wound healing acceleration are highlighted. Furthermore, the potential role of drug-eluting nanofibers in wound management is discussed, and lastly, the economic implications of wounds as well as future perspectives in applying fiber electrospinning in the design of wound dressings and sutures are considered and reported.