Given the current environmental crisis caused by polymers, we have seen that it is mandatory to make changes in the way we produce, consume, and, at the downstream, dispose, and manage polymeric materials after use. For applications where recycling might not be the adequate answer, or landfilling is highly likely to occur, (bio)degradable polymers can play an important role. This appraisal acknowledges the most important synthetic (bio)degradable polymers, noticeably polyesters, and their potential for substitution of traditional persistent ones, in terms of thermal and mechanical properties and more adequate fate in controlled compostable media or instead in natural environments. Special focus is given to bio-based poly(lactic acid), poly(butylene succinate) and furan-based polyesters. Alternatively, the most recent progress and contributions to rendering fossil-based polymers such as poly(caprolactone) and poly(glycolic acid) greener are discussed.