Engineered human tissues created by three-dimensional cell culture of human cells in a hydrogel are becoming emerging model systems for cancer drug discovery and regenerative medicine. Complex functional engineered tissues can also assist in the regeneration, repair, or replacement of human tissues. However, one of the main hurdles for tissue engineering, three-dimensional cell culture, and regenerative medicine is the capability of delivering nutrients and oxygen to cells through the vasculatures. Several studies have investigated different strategies to create a functional vascular system in engineered tissues and organ-on-a-chips. Engineered vasculatures have been used for the studies of angiogenesis, vasculogenesis, as well as drug and cell transports across the endothelium. Moreover, vascular engineering allows the creation of large functional vascular conduits for regenerative medicine purposes. However, there are still many challenges in the creation of vascularized tissue constructs and their biological applications. This review will summarize the latest efforts to create vasculatures and vascularized tissues for cancer research and regenerative medicine.