Worldwide research on electrospinning enabled it as a versatile technique for producing nanofibers with specified physio-chemical characteristics suitable for diverse biomedical applications. In the case of tissue engineering and regenerative medicine, the nanofiber scaffolds' characteristics are custom designed based on the cells and tissues specific needs. This fabrication technique is also innovated for the production of nanofibers with special micro-structure and secondary structure characteristics such as porous fibers, hollow structure, and core-sheath structure. This review attempts to critically and succinctly capture the vast number of developments reported in the literature over the past two decades. We then discuss their applications as scaffolds for induction of cells growth and differentiation or as architecture for being used as graft for tissue engineering. The special nanofibers designed for improving regeneration of several tissues including heart, bone, central nerve system, spinal cord, skin and ocular tissue are introduced. We also discuss the potential of the electrospinning in drug delivery applications, which is a critical factor for cell culture, tissue formation and wound healing applications.