Nowadays, electrospinning has become one of the most versatile, easy, and cost-effective techniques to engineer advanced materials used for many applications, especially in the biomedical and environmental areas. Like the numerous patents around the world, the increasing number of papers witnesses the huge potential of this simple process, and many companies have been emerged during the last years to exploit its innumerable applications. This article presents a critically selected overview of polymers that can be used to produce nanofibers, along with the biomedical applications of the resulting electrospun scaffolds. We have focused on about seven natural and synthetic polymers, but many more can be found in the literature, either as their pristine state or as composites with ceramics, metals, and other polymers. The description of some strategies for nanofiber production, and the characterization used to evaluate their optimization, has been discussed. Finally, several polymers have been recognized as highlights for future work.