Whereas highly porous scaffolds composed of electrospun nanofibers can mimick major features of the extracellular matrix in tissue engineering, they lack the ability to incorporate and release biocompounds (drugs, growth factors) safely in a controlled way. Here, electrospun core-shell fibers (core made from water and aqueous solutions of hydrophilic polymers and the shell from materials with well-defined release mechanisms) offer unique advantages in comparison with those that have helped make porous nanofibrillar scaffolds highly successful in tissue engineering. This review considers the preparation and biofunctionalization of such core-shell fibers as well as applications in various areas, including neural, vascular, cardiac, cartilage and bone tissue engineering, and touches on the topic of clinical trials.