One of the largest fields of application of electrospun materials is the biomedical field, including development of scaffolds for tissue engineering, drug delivery and wound healing. Electrospinning appears as a promising technique in terms of scaffolds composition and architecture, which is the main aspect of this review paper, with a special attention to natural polymers including collagen, fibrinogen, silk fibroin, chitosan, chitin etc. Thanks to the adaptability of the electrospinning process, versatile hybrid, custom tailored structure scaffolds have been reported. The same is achieved due to the vast biomaterials’ processability as well as modifications of the basic electrospinning set-up and its combination with other techniques, simultaneously or by post-processing.