Tissue engineering and regenerative medicine have made great progress in recent decades, as the fields of bioengineering, materials science, and stem cell biology have converged, allowing tissue engineers to replicate the structure and function of various levels of the vascular tree. Nonetheless, the lack of a fully functional vascular system to efficiently supply oxygen and nutrients has hindered the clinical application of bioengineered tissues for transplantation. To investigate vascular biology, drug transport, disease progression, and vascularization of engineered tissues for regenerative medicine, we have analyzed different approaches for designing microvascular networks to create models. This review discusses recent advances in the field of microvascular tissue engineering, explores potential future challenges, and offers methodological recommendations.