With a limited supply of organ donors and available organs for transplantation, the aim of tissue engineering with three-dimensional (3D) bioprinting technology is to construct fully functional and viable tissue and organ replacements for various clinical applications. 3D bioprinting allows for the customization of complex tissue architecture with numerous combinations of materials and printing methods to build different tissue types, and eventually fully functional replacement organs. The main challenge of maintaining 3D printed tissue viability is the inclusion of complex vascular networks for nutrient transport and waste disposal. Rapid development and discoveries in recent years have taken huge strides toward perfecting the incorporation of vascular networks in 3D printed tissue and organs. In this review, we will discuss the latest advancements in fabricating vascularized tissue and organs including novel strategies and materials, and their applications. Our discussion will begin with the exploration of printing vasculature, progress through the current statuses of bioprinting tissue/organoids from bone to muscles to organs, and conclude with relevant applications for in vitro models and drug testing. We will also explore and discuss the current limitations of vascularized tissue engineering and some of the promising future directions this technology may bring.