Biodegradable polymers have become important in different fields of application, where biodegradability and biocompatibility are required. Herein, the melt spinning of biodegradable polymers including poly(lactic acid), poly(butylene succinate), polyhydroxyalkanoate (PHA), poly(e-caprolactone) and their biocomposites is critically reviewed. Biodegradable polymer fibers with added functionalities are in high demand for various applications, including biomedical, textiles, and others. Melt spinning is a suitable technique for the development of biodegradable polymer fibers in a large-scale quantity, and fibers with a high surface area can be obtained with this technique. The processing variables during spinning have a considerable impact on the resulting properties of the fibers. Therefore, in this review, the processing-property relationship in biodegradable polymers, blends, and their composites is provided. The morphological characteristics, load-bearing properties, and the potential application of melt-spun biodegradable fibers in various sectors are also provided.