  This review summarizes more than 100 years of research on spinel compounds,
mainly focusing on the progress in understanding their magnetic, electronic,
and polar properties during the last two decades. Many spinel compounds are
magnetic insulators or semiconductors; however, a number of spinel-type metals
exists including superconductors and some rare examples of d-derived
heavy-fermion compounds. In the early days, they gained importance as
ferrimagnetic or even ferromagnetic insulators with relatively high saturation
magnetization and high ordering temperatures, with magnetite being the first
magnetic mineral known to mankind. However, spinels played an outstanding role
in the development of concepts of magnetism, in testing and verifying the
fundamentals of magnetic exchange, in understanding orbital-ordering and
charge-ordering phenomena. In addition, the A- site as well as the B-site
cations in the spinel structure form lattices prone to strong frustration
effects resulting in exotic ground-state properties. In case the A-site cation
is Jahn-Teller active, additional entanglements of spin and orbital degrees of
freedom appear, which can give rise to a spin-orbital liquid or an orbital
glass state. The B-site cations form a pyrochlore lattice, one of the strongest
contenders of frustration in three dimensions. In addition, in spinels with
both cation lattices carrying magnetic moments, competing magnetic exchange
interactions become important, yielding ground states like the time-honoured
triangular Yafet-Kittel structure. Finally, yet importantly, there exists a
long-standing dispute about the possibility of a polar ground state in spinels,
despite their reported overall cubic symmetry. Indeed, over the years number of
multiferroic spinels were identified.
