  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.
