The first unconventional superconductor was discovered by J.G. Bednorz and K.A. Mueller in 1986. It was a Lanthanum-based cuprate perovskite material with critical temperature of approximately 35K (-238 degrees Celsius). This was well above the highest criticial temperature known at the time (Tc=23K) and thus the new family of materials were called high-temperature superconductors. J.G. Bednorz and K.A. Mueller received the Nobel prize for Physics for this discovery in 1987.
Since then, many other high-temperature superconductors have been synthesized. As early as 1987, superconductivity above 77K, the boiling point of nitrogen, was achieved. This is highly significant from the point of view of the technological applications of superconductivity, because liquid nitrogen is far less expensive than liquid helium, which is required to cool conventional superconductors down to their critical temperature. The current record critical temperature is about Tc=133K (-140°C) at room pressure, and somewhat higher critical temperatures can be achieved at high pressure. Nevertheless at present it is considered unlikely that cuprate perovskite materials will achieve room-temperature superconducitivty.
On the other hand, in recent years other unconventional superconductors have been discovered. These include some that do not superconduct at high temperatures, such as the strontium-ruthenate oxide compounds, but that, like the high-temperature superconductors, are unconventional in other ways (for example, the origin of the attractive force leading to the formation of Cooper pairs may be different from the one postulated in BCS theory). In addition to this, supeconductors that have unusually high values of Tc but that are not cuprate perovskites have been discovered. Some of them may be extreme examples of conventional superconductors (this is suspected of magnesium diboride, MgB2, with Tc=39K). Others display more unconventional features.