Collagen and gelatin are essential natural biopolymers commonly utilized in biomaterials and tissue engineering because of their excellent physicochemical and biocompatibility properties. They can be used either in combination with other biomacromolecules or particles or even exclusively for the enhancement of bone regeneration or for the development of biomimetic scaffolds. Collagen or gelatin derivatives can be transformed into nanofibrous materials with porous micro- or nanostructures and superior mechanical properties and biocompatibility using electrospinning technology. Specific attention was recently paid to electrospun mats of such biopolymers, due to their high ratio of surface area to volume, as well as their biocompatibility, biodegradability, and low immunogenicity. The fiber mats with submicro- and nanometer scale can replicate the extracellular matrix structure of human tissues and organs, making them highly suitable for use in tissue engineering due to their exceptional bioaffinity. The drawbacks may include rapid degradation and complete dissolution in aqueous media. The use of gelatin/collagen electrospun nanofibers in this form is thus greatly restricted for biomedicine. Therefore, the cross-linking of these fibers is necessary for controlling their aqueous solubility. This led to enhanced biological characteristics of the fibers, rendering them excellent options for various biomedical uses. The objective of this review is to highlight the key research related to the electrospinning of collagen and gelatin, as well as their applications in the biomedical field. The review features a detailed examination of the electrospinning fiber mats, showcasing their varying structures and performances resulting from diverse solvents, electrospinning processes, and cross-linking methods. Judiciously selected examples from literature will be presented to demonstrate major advantages of such biofibers. The current developments and difficulties in this area of research are also being addressed.