There is a general increase in the number of patients with non-healing skin wounds, imposing a huge social and economic burden on patients and healthcare systems. Severe skin injury is an important clinical challenge. There is a lack of skin donors, and skin defects and scarring after surgery can lead to impaired skin function and skin integrity. Researchers worldwide have made great efforts to create human skin organs but are limited by the lack of key biological structural features of the skin. Tissue engineering repairs damaged tissue by incorporating cells into biocompatible and biodegradable porous scaffolds. Skin tissue engineered scaffolds not only have appropriate physical and mechanical properties but also exhibit skin-like surface topography and microstructure, which can promote cell adhesion, proliferation, and differentiation. At present, skin tissue engineering scaffolds are being developed into clinical applications that can overcome the limitations of skin transplantation, promote the process of wound healing, and repair skin tissue damage. This provides an effective therapeutic option for the management of patients with skin lesions. This paper reviews the structure and function of skin tissue and the process of wound healing, and summarizes the materials and manufacturing methods used to fabricate skin tissue engineering scaffolds. Next, the design considerations of skin tissue engineering scaffolds are discussed. An extensive review of skin scaffolds and clinically approved scaffold materials is presented. Lastly, some important challenges in the construction of skin tissue engineering scaffolds are presented.