Thin-film transistors (TFTs) fabricated using amorphous oxide semiconductors (AOS) exhibit good electron mobility (5 to > 50cm^2/V·s), they are transparent, and they can be processed at low temperatures. These new materials show a great promise for high-performance large-area electronics applications such as flexible electronics, transparent electronics, and analog current drivers for organic light-emitting diode displays. Before any of these applications can be commercialized, however, a strong understanding of the stability and reliability of AOS TFTs is needed. The purpose of this paper is to provide a comprehensive review and summary of the recently emerging work on the stability and reliability of AOS TFTs with respect to illumination, bias stress, ambient effects, surface passivation, mechanical stress, and defects, as well as to point out areas for future work. An overview of the TFT operation and expected reliability concerns as well as a brief summary of the instabilities in the well-known Si3N4/a-Si:H system is also included.