Bioprinting is a promising technology to fabricate design-specific tissue constructs due to its ability to create complex, heterocellular structures with anatomical precision. Bioprinting enables the deposition of various biologics including growth factors, cells, genes, neo-tissues and extra-cellular matrix-like hydrogels. Benefits of bioprinting have started to make a mark in the fields of tissue engineering, regenerative medicine and pharmaceutics. Specifically, in the field of tissue engineering, the creation of vascularized tissue constructs has remained a principal challenge till date. However, given the myriad advantages over other biofabrication methods, it becomes organic to expect that bioprinting can provide a viable solution for the vascularization problem, and facilitate the clinical translation of tissue engineered constructs. This article provides a comprehensive account of bioprinting of vascular and vascularized tissue constructs. The review is structured as introducing the scope of bioprinting in tissue engineering applications, key vascular anatomical features and then a thorough coverage of 3D bioprinting using extrusion-, droplet- and laser-based bioprinting for fabrication of vascular tissue constructs. The review then provides the reader with the use of bioprinting for obtaining thick vascularized tissues using sacrificial bioink materials. Current challenges are discussed, a comparative evaluation of different bioprinting modalities is presented and future prospects are provided to the reader.Biofabrication of living tissues and organs at the clinically-relevant volumes vitally depends on the integration of vascular network. Despite the great progress in traditional biofabrication approaches, building perfusable hierarchical vascular network is a major challenge. Bioprinting is an emerging technology to fabricate design-specific tissue constructs due to its ability to create complex, heterocellular structures with anatomical precision, which holds a great promise in fabrication of vascular or vascularized tissues for transplantation use. Although a great progress has recently been made on building perfusable tissues and branched vascular network, a comprehensive review on the state-of-the-art in vascular and vascularized tissue bioprinting has not reported so far. This contribution is thus significant because it discusses the use of three major bioprinting modalities in vascular tissue biofabrication for the first time in the literature and compares their strengths and limitations in details. Moreover, the use of scaffold-based and scaffold-free bioprinting is expounded within the domain of vascular tissue fabrication.