Organ transplantation is the most effective therapy for end-stage organ failure. However, the demand for life-saving organ transplants far exceeds the supply of available organs owing to organ shortage. To address this problem, tissue engineering has offered potential strategies for in vitro construction of organs as medical and clinical applications. However, tissue-engineered organs are difficult to construct owing to the lack of functional vascular networks because avascular organs lead to tissue dysfunctions, such as hypoxia and clot formation. Therefore, establishing functional vascular networks is required for the construction and maintenance of organs in terms of morphology and function. Recent advances in tissue engineering have allowed the in vitro construction of a wide range of functional vascular networks, ranging from microvessels to organ-scale vascular networks, using self-organization and pre-designed approaches. In particular, various new models have been developed utilizing microfluidics, 3D bioprinting, and organ decellularization. These models have enabled the in vitro recapitulation of key features of physiological vascular networks, such as morphology (e.g., network formation, luminal structure, and perivascular cell coverage) and function (e.g., barrier and antithrombogenic functions). In this review, we summarize the progress and challenges in vascular tissue engineering based on two distinct categories: self-organization and pre-designed approaches. In addition, the advantages and limitations of these models are highlighted, and future perspectives are discussed. These models will provide useful insights for the construction of vascularized functional tissues and organs and can contribute to development in tissue engineering and regenerative medicine.