Developments in micro- and nanofluidic technologies have led to new kinds of cell culture and screening systems that are collectively termed organ-on-a-chip systems. Organ-on-a-chip systems are in vitro microfabricated devices that mimic dynamic interactions of in vivo microenvironments. In addition to existing two-dimensional and three-dimensional tissues, organ-on-a-chip systems can mimic the biomechanical and biochemical microenvironments of in vivo tissues as well as the interactional effects of the microenvironments on cell and tissue functions. Owing to those features, organ-on-a-chip systems have become excellent platforms for drug screening and delivery tests. In this review, specific examples of organ-on-a-chip devices and their applications in tissue engineering and drug delivery tests are presented. The utility and performance of state-of-the-art organ-on-a-chip systems, including lung-on-a-chip, heart-on-a-chip, vessel-on-a-chip, liver-on-a-chip, and tumor-on-a-chip, are also covered in this review. Limitations of conventional systems, basic fabrication processes for organ-on-a-chip devices, and future prospects of organ-on-a-chip systems are discussed.