Designing biomimetic scaffolds with the ability for tissue regeneration is an important strategy in tissue engineering. The unique biocompatibility and biodegradability of biopolymers make them an excellent choice for tissue engineering and regenerative medicine. Biopolymers, which have a wide range of properties, can be obtained from different natural sources. Depending on the target tissue, biopolymers can be engineered to guarantee a specific function. We review the different types of biopolymer-based composites and their interactions with specific cells and tissues. Specific cellular mechanisms in tissue regeneration are also reviewed to elucidate the effects of biopolymers on controlling cellular mechanisms given their advantages and challenging aspects. Furthermore, the modifications required to mimic the properties of neural, cardiac, bone, and skin tissues are discussed. Studies on biopolymer-based composites utilization in tissue engineering require additional improvement and many crucial challenges are yet to be overcome.