The biofabrication of biomimetic scaffolds for tissue engineering applications is a field in continuous expansion. Of particular interest, nanofibrous scaffolds can mimic the mechanical and structural properties (e.g., collagen fibers) of the natural extracellular matrix (ECM) and have shown high potential in tissue engineering and regenerative medicine. This review presents a general overview on nanofiber fabrication, with a specific focus on the design and application of electrospun nanofibrous scaffolds for vascular regeneration. The main nanofiber fabrication approaches, including self-assembly, thermally induced phase separation, and electrospinning are described. We also address nanofibrous scaffold design, including nanofiber structuring and surface functionalization, to improve scaffolds' properties. Scaffolds for vascular regeneration with enhanced functional properties, given by providing cells with structural or bioactive cues, are discussed. Finally, current in vivo evaluation strategies of these nanofibrous scaffolds are introduced as the final step, before their potential application in clinical vascular tissue engineering can be further assessed.