Cardiovascular defects, injuries, and degenerative diseases often require surgical intervention and the use of implantable replacement material and conduits. Traditional vascular grafts made of synthetic polymers, animal and cadaveric tissues, or autologous vasculature have been utilized for almost a century with well-characterized outcomes, leaving areas of unmet need for the patients in terms of durability and long-term patency, susceptibility to infection, immunogenicity associated with the risk of rejection, and inflammation and mechanical failure. Research to address these limitations is exploring avenues as diverse as gene therapy, cell therapy, cell reprogramming, and bioengineering of human tissue and replacement organs. Tissue-engineered vascular conduits, either with viable autologous cells or decellularized, are the forefront of technology in cardiovascular reconstruction and offer many benefits over traditional graft materials, particularly in the potential for the implanted material to be adopted and remodeled into host tissue and thus offer safer, more durable performance. This review discusses the key advances and future directions in the field of surgical vascular repair, replacement, and reconstruction, with a focus on the challenges and expected benefits of bioengineering human tissues and blood vessels.