High Temperature Superconductivity
The first 'high-temperature' (defined as above –196.2°C) 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.
(Superconductivity is a state in which a substance completely loses all electrical resistance - in other words an electrical current can flow through it, for ever, without any losses or heat generation)
Superconductivity itself, in metals, had been previously found in 1911, using mercury cooled by (the recently-discovered) liquid helium. But it took until 1957 for the phenomenon to be explained via the BCS theory. The theory, however, is only applicable up to around 30 or so degrees above absolute zero.
'High temperature' superconducting remains completely unexplained.
It is one of the biggest challenges of modern times […] in physics. Something like fifteen Nobel laureates have written articles on high temperature superconductivity. I for one know of no other sub-field in physics that has energised or sparked the imagination of so many of the greatest minds."
Source : Nigel Hussey, professor of experimental condensed matter physics at the university of Bristol, UK, speaking on BBC's In Our Time, podcast, Superconductivity Jan 2023.
It's now widely accepted that an explanation will not be found using models which rely on the (highly predictable) behaviour of single (or paired) electrons. Instead current thinking leans towards as-yet-unknown 'emergent' properties of 'coherent' clouds of quantum-entangled electrons.
Recent progress in the proposed theoretical models can be found at Wikipedia
Room Temperature superconductors :
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 superconductivity (at around 15°C) is featured in Nature volume 586, pages 373–377(2020) The material tested was H2S + H2 - Unfortunately, for practical applications, the effect only occurs at the extreme pressures of 267 ± 10 gigapascals - only achievable in a diamond anvil-cell.
IMPORTANT NOTE : This paper cited above has now been 'retracted' by Nature due to "concerns regarding the manner in which the data in this paper have been processed and interpreted."
Update July 2023 : A report of room temperature superconductivity - at normal atmospheric pressure - has been uploaded to arXiv
A joint Korean / US research team say their newly developed crystalline compound - a Cu-doped lead apatite (LA) - has near zero electrical resistance.
IMPORTANT NOTE This is not a peer-reviewed paper, and the report has not been confirmed by other research teams. Many superconductivity researchers are currently sceptical about the the published results.
Update Aug 2023 : A research team from the University of Illinois Urbana-Champaign, US, have announced experimental confirmation of a 'plasmon' particle called "Pines' Demon".
The 'Demon' particles were predicted nearly 70 years ago by US theoretical physicist David Pines. He proposed that the composite particles could form when electrons in different energy bands (in metals) vibrated out-of-phase, leading tp modulations in the band occupancy. Although formed from electrons, the particles would be mass-less, neutral, and would not interact with light.
They have not been detected experimentally before, perhaps because of their extreme lack of interaction.
Various research groups have previously conjectured that such particles could offer a possible explanation for high temperature superconductivity. (see paper linked below for references). More research, however, is needed to confirm or refute the 'Demon's' possible role.
See Nature 
Aug 2023
Update Nov. 2023 :
Another recent paper regarding possible room temperature superconductivity, again published in the journal Nature has also been retracted from the journal .
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