The development of biodegradable and biocompatible materials to replace non-biodegradable synthetic polymers has received significant interest in biomedical applications. Amongst all biopolymers, polybutylene succinate (PBS) possesses excellent properties, such as biocompatibility, easy processability, biodegradability, non-toxic, good mechanical and thermal properties, which makes it a suitable candidate to replace non-biodegradable synthetic polymers in biomedical applications. This review provides recent advancements, trends, and challenges for the utilization of polybutylene succinate (PBS) and PBS-based materials for biomedical applications, viz. drug delivery, tissue engineering, and biomedical devices. In addition, the design of medical devices from PBS using 3D and 4D printing is also presented. The market analysis, comparison of virgin PBS and synthetic polymer properties, as well as the end of service life, are summarized briefly. The current status, challenges, and future directions are also discussed.