Angiogenesis (or the development of new blood vessels) is a key event in tissue engineering and regenerative medicine; thus, a number of biomaterials have been developed and combined with stem cells and/or bioactive molecules to produce three-dimensional (3D) pro-angiogenic constructs. Among the various biomaterials, electrospun nanofibrous scaffolds offer great opportunities for pro-angiogenic approaches in tissue repair and regeneration. Nanofibers made of natural and synthetic polymers are often used to incorporate bioactive components (e.g., bioactive glasses (BGs)) and load biomolecules (e.g., vascular endothelial growth factor (VEGF)) that exert pro-angiogenic activity. Furthermore, seeding of specific types of stem cells (e.g., endothelial progenitor cells) onto nanofibrous scaffolds is considered as a valuable alternative for inducing angiogenesis. The effectiveness of these strategies has been extensively examined both in vitro and in vivo and the outcomes have shown promise in the reconstruction of hard and soft tissues (mainly bone and skin, respectively). However, the translational of electrospun scaffolds with pro-angiogenic molecules or cells is only at its beginning, requiring more research to prove their usefulness in the repair and regeneration of other highly-vascularized vital tissues and organs. This review will cover the latest progress in designing and developing pro-angiogenic electrospun nanofibers and evaluate their usefulness in a tissue engineering and regenerative medicine setting.