Conventional superconductivity, as used in this review, refers to electron-phonon coupled superconducting electron-pairs described by BCS theory. Unconventional superconductivity refers to superconductors where the Cooper pairs are not bound together by phonon exchange but instead by exchange of some other kind, e. g. spin fluctuations in a superconductor with magnetic order either coexistent or nearby in the phase diagram. Such unconventional superconductivity has been known experimentally since heavy fermion CeCu2Si2, with its strongly correlated 4f electrons, was discovered to superconduct below 0.6 K in 1979. Since the discovery of unconventional superconductivity in the layered cuprates in 1986, the study of these materials saw Tc jump to 164 K by 1994. Further progress in high temperature superconductivity would be aided by understanding the cause of such unconventional pairing. This review compares the fundamental properties of 9 unconventional superconducting classes of materials - from 4f-electron heavy fermions to organic superconductors to classes where only three known members exist to the cuprates with over 200 examples, with the hope that common features will emerge to help theory explain (and predict!) these phenomena. In addition, three new emerging classes of superconductors (topological, interfacial [e. g. FeSe on SrTiO3], and H2S under high pressure) are briefly covered, even though their conventionality is not yet fully determined.
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