Nowadays the open and the patent literatures propose a large number of polymers whose main chains can be degraded usefully. Among these degradable polymers, aliphatic polyester-based polymeric structures are receiving special attention because they are all more or less sensitive to hydrolytic degradation, a feature of interest when compared with the fact that living systems function in aqueous media. Only some of these aliphatic polyesters are enzymatically degradable. A smaller number is biodegradable, and an even more limited number is biorecyclable. To be of practical interest, a degradable polymer must fulfill many requirements that depend very much on the targeted application, on the considered living system, and on living conditions. It is shown that aliphatic polyester structures made of repeating units that can generate metabolites upon degradation or biodegradation like poly(beta-hydroxy alkanoate)s and poly(alpha-hydroxy alkanoate)s are of special interest. Their main characteristics are confronted to the specifications required by various potential sectors of applications, namely, surgery, pharmacology, and the environment. It is shown that degradation, bioresorption, and biorecycling that are targets when one wants to respect living systems are also drastic limiting factors when one wants to achieve a device of practical interest. Finding a universal polymer that would be the source of all the polymeric biomaterials needed to work in contact with living organisms of the various life kingdoms and respect them remains a dream. On the other hand, finding one polymeric structure than can fulfill the requirements of one niche application remains a big issue.