Biodegradable materials are widely used in the biomedical field because there is no postoperative surgery after implantation. Widely used synthetic biodegradable materials are polyesters, especially those used in tissue engineering. Advances in the tissue engineering field have brought much attention in terms of scaffold fabrication, such as with biodegradable polyester nanofibers. The rationale for using nanofibers for tissue engineering is that the nonwoven polymeric meshwork is a close representation of the nanoscale protein fiber meshwork in native extracellular matrix (ECM). Electrospinning technique is a promising way to fabricate controllable continuous nanofiber scaffold mimicking the ECM structure. Electrospun nanofibers provide high surface-to-volume ratio and high porosity as a promising scaffold for tissue engineering. Because the degradation behaviors of scaffolds significantly affect new tissue regeneration, the degradation of the material becomes one of the crucial factors when considering using polyester nanofibers as scaffolds in tissue engineering. In this review paper, we focus on the degradation studies of several bioresorbable polyester nanofibrous scaffolds used in tissue engineering. The degradable properties of nanofibers were compared with the corresponding degradable materials in macroscale. The factors that might affect the degradation behaviors were analyzed.