Significant advances in biomaterials, stem cell biology, and microscale technologies have enabled the fabrication of biologically relevant tissues and organs. Such tissues and organs, referred to as organ-on-a-chip (OOC) platforms, have emerged as a powerful tool in tissue analysis and disease modeling for biological and pharmacological applications. A variety of biomaterials are used in tissue fabrication providing multiple biological, structural, and mechanical cues in the regulation of cell behavior and tissue morphogenesis. Cells derived from humans enable the fabrication of personalized OOC platforms. Microscale technologies are specifically helpful in providing physiological microenvironments for tissues and organs. In this review, biomaterials, cells, and microscale technologies are described as essential components to construct OOC platforms. The latest developments in OOC platforms (e.g., liver, skeletal muscle, cardiac, cancer, lung, skin, bone, and brain) are then discussed as functional tools in simulating human physiology and metabolism. Future perspectives and major challenges in the development of OOC platforms toward accelerating clinical studies of drug discovery are finally highlighted.