The development of advanced engineering strategies to create realistic model systems that facilitate the exploration of the profound complexity of biological tissues is essential for enhancing our understanding of human health and disease. To this end, we present engineering approaches to create stem cell-derived three-dimensional (3D) miniature models of human organs on a chip that mimic the physiology and function of living human tissues. First, we demonstrate the advanced application of the organs-on-a-chip by presenting multiple microengineered tissue model systems: i) self-assembled and perfusable microvascular beds and vascularized 3D human microtissues and ii) blood-retina barrier of the human retina. Using our vascular engineering strategy, we further demonstrate advanced capabilities of our microengineered models to incorporate patient-derived induced pluripotent stem cells to recreate fat and retina tissues. These model systems were used to investigate key pathological features of white adipose tissue and age-related macular degeneration in patients and showed their potential to revolutionize drug testing and therapeutic target identification. Next, we describe a novel approach to synergistically combine organs-on-a-chip technology with organoids to create human tissue models with increased biological and structural complexity. Specifically, we introduce a novel approach to enhance organogenesis in a dish through geometric engineering of organoid culture. Our approach allows sustained organoid development over prolonged culture period enabling accelerated production intestinal organoids with increased structural and functional maturity. Additionally, we demonstrate successful replication of key pathological features of inflammatory bowel disease (IBD) and production of vascularized, perfusable IBD patient enteroids for the study of immune responses in IBD. Finally, we present an alveologenesis-on-a-chip model that enables simulation of the fetal breathing movement-induced mechanical forces. The proposed system was then utilized to understand how forces, particularly those generated by fetal breathing movements, affect the differentiation and maturation of alveolar epithelial cells during organoid development. Overall, the work presented in this dissertation presents several key innovations that represent significant advances from the existing approaches. We believe that our approach to synergistically leverage stem cells and organoids with organ-on-a-chip technology represents a significant contribution to the biomedicine community and will improve our fundamental knowledge of human health and disease.