This article provides a thorough overview of the available resorbable biomaterials appropriate for producing replacements for damaged tissues. In addition, their various properties and application possibilities are discussed as well. Biomaterials are fundamental components in tissue engineering (TE) of scaffolds and play a critical role. They need to exhibit biocompatibility, bioactivity, biodegradability, and non-toxicity, to ensure their ability to function effectively with an appropriate host response. With ongoing research and advancements in biomaterials for medical implants, the objective of this review is to explore recently developed implantable scaffold materials for various tissues. The categorization of biomaterials in this paper includes fossil-based materials (e.g., PCL, PVA, PU, PEG, and PPF), natural or bio-based materials (e.g., HA, PLA, PHB, PHBV, chitosan, fibrin, collagen, starch, and hydrogels), and hybrid biomaterials (e.g., PCL/PLA, PCL/PEG, PLA/PEG, PLA/PHB PCL/collagen, PCL/chitosan, PCL/starch, and PLA/bioceramics). The application of these biomaterials in both hard and soft TE is considered, with a particular focus on their physicochemical, mechanical, and biological properties. Furthermore, the interactions between scaffolds and the host immune system in the context of scaffold-driven tissue regeneration are discussed. Additionally, the article briefly mentions the concept of in situ TE, which leverages the self-renewal capacities of affected tissues and highlights the crucial role played by biopolymer-based scaffolds in this strategy.