Synthetic biodegradable polymers are of great value for the preparation of implants that are required to reside only temporarily in the body. The use of biodegradable polymers obviates the need for a second surgery to remove the implant, which is the case when a nondegradable implant is used. After implantation in the body, biomedical devices may be subjected to degradation and erosion. Understanding the mechanisms of these processes is essential for the development of biomedical devices or implants with a specific function, for example, scaffolds for tissue-engineering applications. For the engineering and regeneration of soft tissues (e.g., blood vessels, cardiac muscle and peripheral nerves), biodegradable polymers are needed that are flexible and elastic. The scaffolds prepared from these polymers should have tuneable degradation properties and should perform well under long-term cyclic deformation conditions. The required polymers, which are either physically or chemically crosslinked biodegradable elastomers, are reviewed in this article.