The copper oxides (cuprates) stand out in their high T c values, 135 K, and are only family that achieved T c exceeding 100 K. Thanks to the enormous research efforts over 28 years, understanding of the cuprates has remarkably advanced, though some important issues remain under hot debate. A consensus which has been achieved is that primarily repulsive electron-electron interactions lead to superconductivity with such high T c as well as to unconventional d-wave pairing. Superconductivity emerges upon doping into the Mott insulating CuO 2 planes. The temperature-doping phase diagram of the cuprates is extremely complex with several orders or incipient orders showing up. Among the known orders, the pseudogap order encompasses a wide doping-temperature region and appears to be home to the superconducting order and its fluctuation at very high temperatures. In contrast to these generic features, T c values of the cuprates are strongly material dependent. A few parameters, such as disorder and multilayer, have been identified that control the wide-spread T c . T c is actually enhanced in some cuprates by reducing disorder in the crystal, and there is a possibility of T c enhancement to 150 K or even higher by optimizing multilayer structure.