The development of bio-based and biodegradable plastics has emerged as a topic of interest for creating novel materials capable of replacing conventional plastics in various sustainable applications such as textiles. Polyhydroxyalkanoates (PHAs), a well-known class of naturally occurring thermoplastic polyesters, have drawn increasing interest as sustainable alternatives to synthetic thermoplastics in many textile applications because of their non-toxic nature and biodegradability in the ambient environment. However, the PHAs’ complex crystallization behavior and the resulting intricate processing procedures remain a challenge. This work provides a comprehensive overview of the advancements in the realm of biodegradable textile manufacturing, namely melt-spun fibers and melt-blown nonwovens. It focuses on discussing the developments in processing, their challenges, process parameters, and product characteristics. It also highlights the challenges encountered in the commercialization of these textile applications and addresses the potential solutions.