The first high-temperature superconductor was discovered in 1986 by IBM researchers Karl Müller and Johannes Bednorz, for which they were awarded the Nobel Prize in Physics in 1987. Since then many other materials with even higher superconducting temperatures have been identified. Although superconductivity at very low temperatures has been understood since 1957 via BCS theory - 'high temperature' superconducting is still unexplained.
Recent progress in the proposed theoretical models can be found at Wikipedia
Since there is no agreed explanation as yet, room-temperature superconductors (which would have the capacity to totally revolutionise human technology) are not ruled out.
Update Oct 2020 : A report of room-temperature supconductivity (at around 15C) is reported inNature volume 586, pages373–377(2020)
The material tested was H2S + H2 - Unfortunately for practical applications, the effect only occurs at pressures of 267 ± 10 gigapascals - only achievable in a diamond anvil-cell.
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