Biodegradable polymeric biomaterials offer a significant advantage in disposable or fast-consuming products in medical applications. Poly(3-hydroxybutyrate--3-hydroxyvalerate) (PHBV) is an example of a polyhydroxyalkanoate (PHA), i.e., one group of natural polyesters that are byproducts of reactions taking place in microorganisms in conditions with an excess carbon source. PHA polymers are a promising material for the production of everyday materials and biomedical applications. Due to the high number of monomers in the group, PHAs permit modifications enabling the production of copolymers of different compositions and with different proportions of individual monomers. In order to change and improve the properties of polymer fibers, PHAs are combined with either other natural and synthetic polymers or additives of inorganic phases. Importantly, electrospun PHBV fibers and mats showed an enormous potential in both the medical field (tissue engineering scaffolds, plasters, wound healing, drug delivery systems) and industrial applications (filter systems, food packaging). This Review summarizes the current state of the art in processing PHBV, especially by electrospinning, its degradation processes, and biocompatibility studies, starting from a general introduction to the PHA group of polymers.