There is a growing demand for off-the-shelf tissue engineered vascular grafts (TEVGs) for the replacement or bypass of damaged arteries in various cardiovascular diseases. Scaffolds from the decellularized tissue skeletons to biopolymers and biodegradable synthetic polymers have been used for fabricating TEVGs. However, several issues have not yet been resolved, which include the inability to mimic the mechanical properties of native tissues, and the ability for long-term patency and growth required for in vivo function. Electrospinning is a popular technique for the production of scaffolds that has the potential to address these issues. However, its application to human TEVGs has not yet been achieved. This review provides an overview of tubular scaffolds that have been prepared by electrospinning with potential for TEVG applications.