The human circulatory system is divided into two complementary and different systems, the cardiovascular and the lymphatic system. The cardiovascular system is mainly concerned with providing nutrients to the body via blood and transporting wastes away from the tissues to be released from the body. The lymphatic system focuses on the transport of fluid, cells, and lipid from interstitial tissue spaces to lymph nodes and, ultimately, to the cardiovascular system, as well as helps coordinate interstitial fluid and lipid homeostasis and immune responses. In addition to having distinct structures from each other, each system also has organ-specific variations throughout the body and both systems play important roles in maintaining homeostasis. Dysfunction of either system leads to devastating and potentially fatal diseases, warranting accurate models of both blood and lymphatic vessels for better studies. As these models also require physiological flow (luminal and interstitial), extracellular matrix conditions, dimensionality, chemotactic biochemical gradient, and stiffness, to better reflect in vivo, three dimensional (3D) microfluidic (on-a-chip) devices are promising platforms to model human physiology and pathology. In this review, we discuss the heterogeneity of both blood and lymphatic vessels, as well as current in vitro models. We, then, explore the organ-specific features of each system with examples in the gut and the brain and the implications of dysfunction of either vasculature in these organs. We close the review with discussions on current in vitro models for specific diseases with an emphasis on on-chip techniques.