Recent advancements in the bioinks and three-dimensional (3D) bioprinting methods used to fabricate vascular constructs are summarized herein. Critical biomechanical properties required to fabricate an ideal vascular graft are highlighted, as well as various testing methods have been outlined to evaluate the bio-fabricated grafts as per the Food and Drug Administration (FDA) and International Organization for Standardization (ISO) guidelines. Occlusive artery disease and cardiovascular disease are the major causes of death globally. These diseases are caused by the blockage in the arteries, which results in a decreased blood flow to the tissues of major organs in the body, such as the heart. Bypass surgery is often performed using a vascular graft to re-route the blood flow. Autologous grafts represent a gold standard for such bypass surgeries; however, these grafts may be unavailable due to the previous harvesting or possess a poor quality. Synthetic grafts serve well for medium to large-sized vessels, but they fail when used to replace small-diameter vessels, generally smaller than 6 mm. Various tissue engineering approaches have been used to address the urgent need for vascular graft that can withstand hemodynamic blood pressure and has the ability to grow and remodel. Among these approaches, 3D bioprinting offers an attractive solution to construct patient-specific vessel grafts with layered biomimetic structures.