Despite all the efforts that have been done up to now, the currently available wound dressings are still unable to fully re-establish all the structural and functional properties of the native skin. To overcome this situation, researchers from the tissue engineering area have been developing new wound dressings (hydrogels, films, sponges, membranes) aiming to mimic all the features of native skin. Among them, asymmetric membranes emerged as a promising solution since they reproduce both epidermal and dermal skin layers. Wet or dry/wet phase inversion, scCO-assisted phase inversion, and electrospinning have been the most used techniques to produce such a type of membranes. Among them, the electrospinning technique, due to its versatility, allows the development of multifunctional dressings, using natural and/or synthetic polymers, which resemble the extracellular matrix of native skin as well as address the specific requirements of each skin layer. Moreover, various therapeutic or antimicrobial agents have been loaded within nanofibers to further improve the wound healing performance of these membranes. This review article provides an overview of the application of asymmetric electrospun membranes as wound dressings displaying antibacterial activity and as delivery systems of biomolecules that act as wound healing enhancers.