The present status and recent research results on amorphous oxide semiconductors (AOSs) and their thin-film transistors (TFTs) are reviewed. AOSs represented by amorphous In-Ga-Zn-O (a-IGZO) are expected to be the channel material of TFTs in next-generation flat-panel displays because a-IGZO TFTs satisfy almost all the requirements for organic light-emitting-diode displays, large and fast liquid crystal and three-dimensional (3D) displays, which cannot be satisfied using conventional silicon and organic TFTs. The major insights of this review are summarized as follows. (i) Most device issues, such as uniformity, long-term stability against bias stress and TFT performance, are solved for a-IGZO TFTs. (ii) A sixth-generation (6G) process is demonstrated for 32″ and 37″ displays. (iii) An 8G sputtering apparatus and a sputtering target have been developed. (iv) The important effect of deep subgap states on illumination instability is revealed. (v) Illumination instability under negative bias has been intensively studied, and some mechanisms are proposed. (vi) Degradation mechanisms are classified into back-channel effects, the creation of traps at an interface and in the gate insulator, and the creation of donor states in annealed a-IGZO TFTs by the Joule heating; the creation of bulk defects should also be considered in the case of unannealed a-IGZO TFTs. (vii) Dense passivation layers improve the stability and photoresponse and are necessary for practical applications. (viii) Sufficient knowledge of electronic structures and electron transport in a-IGZO has been accumulated to construct device simulation models.