The global transplantation market size was valued at USD 8.4 billion in 2020 and is expected to grow at a compound annual growth rate of 11.5% over the forecast period. The increasing demand for tissue transplantation has inspired researchers to find alternative approaches for making artificial tissues and organs function. The unique physicochemical and biological properties of biopolymers and the attractive structural characteristics of aerogels such as extremely high porosity, ultra low-density, and high surface area make combining these materials of great interest in tissue scaffolding and regenerative medicine applications. Numerous biopolymer aerogel scaffolds have been used to regenerate skin, cartilage, bone, and even heart valves and blood vessels by growing desired cells together with the growth factor in tissue engineering scaffolds. This review focuses on the principle of tissue engineering and regenerative medicine and the role of biopolymer aerogel scaffolds in this field, going through the properties and the desirable characteristics of biopolymers and biopolymer tissue scaffolds in tissue engineering applications. The recent advances of using biopolymer aerogel scaffolds in the regeneration of skin, cartilage, bone, and heart valves are also discussed in the present review. Finally, we highlight the main challenges of biopolymer-based scaffolds and the prospects of using these materials in regenerative medicine.