Polybutylene succinate (PBS) and other succinic (co)polyesters are biodegradable polymers with favorable mechanical and thermal properties that find use in many applications. Due to environmental concerns, polymers based on succinic acid (SA) have been gaining attention, as SA can be produced through biotechnological processes. Thus, this review aims to highlight the synthesis and characteristics of PBS and other succinic copolyesters, with emphasis in the works employing metallic catalysts and enzymes. In addition, the modification of the macromolecular structure by copolymerization or postpolymerization is also discussed. Currently, metallic catalysts are normally used in the synthesis of these materials, under conditions of high temperatures, which can favor the occurrence of thermal degradation, increasing the dispersion of chain length distributions. Moreover, the incrustation of metallic catalysts in polymeric materials makes their application in biomedical products difficult, due to toxicity requirements. In this context, enzymatic catalysis is gaining ground, offering milder synthesis temperatures, high selectivity, and uniformity of synthesized products. This biotechnological route can substitute oligomerization processes with metallic catalysis in future industrial processes, producing materials free from metallic contamination. In addition to production by catalytic routes, trends for future applications of succinic (co)polyesters are presented, with emphasis on the value‐added materials sectors.