In recent years, 3D printing techniques experience a growing interest in several sectors, including the biomedical one. Their main advantage resides in the possibility to obtain complex and personalized structures in a cost-effective way impossible to achieve with traditional production methods. This is especially true for fused deposition modeling (FDM), one of the most diffused 3D printing methods. The easy customization of the final products' geometry, composition, and physicochemical properties is particularly interesting for the increasingly personalized approach adopted in modern medicine. Thermoplastic polymers are the preferred choice for FDM applications, and a wide selection of biocompatible and biodegradable materials is available to this aim. Moreover, these polymers can also be easily modified before and after printing to better suit the body environment and the mechanical properties of biological tissues. This review focuses on the use of thermoplastic aliphatic polyesters for FDM applications in the biomedical field. In detail, the use of poly(ε-caprolactone), poly(lactic acid), poly(lactic-co-glycolic acid), poly(hydroxyalkanoate)s, thermoplastic poly(ester urethane)s, and their blends is thoroughly surveyed, with particular attention to their main features, applicability, and workability. The state-of-the-art is presented and current challenges in integrating the additive manufacturing technology in the medical practice are discussed.