  Mott insulators with large and active (or multiflavor) local Hilbert spaces
widely occur in quantum materials and ultracold atomic systems, and are dubbed
"multiflavor Mott insulators". For these multiflavored Mott insulating
materials, the spin-only description with the quadratic spin interactions is
often insufficient to capture the major physical processes. In the situation
with active orbitals, the Kugel-Khomskii superexchange model was then proposed.
We briefly review this historical model and discuss the modern developments
beyond the original spin-orbital context. These include and are not restricted
to the $4d$/$5d$ transition metal compounds with the spin-orbit-entangled
$J=3/2$ quadruplets, the rare-earth magnets with two weakly-separated crystal
field doublets, breathing magnets and/or the cluster and molecular magnets, et
al. We explain the microscopic origin of the emergent Kugel-Khomskii physics in
each realization with some emphasis on the $J=3/2$ quadruplets, and refer the
candidate multiflavor Mott insulators as "$J=3/2$ Mott insulators". For the
ultracold atoms, we review the multiflavor Mott insulator realization with the
ultracold alkaline and alkaline-earth atoms on the optical lattices. Despite a
large local Hilbert space from the atomic hyperfine spin states, the system
could naturally realize a large symmetry group such as the Sp($N$) and SU($N$)
symmetries. These ultracold atomic systems lie in the large-$N$ regime of these
symmetry groups and are characterized by strong quantum fluctuations. The
Kugel-Khomskii physics and the exotic quantum ground states with the
"baryon-like" physics can appear in various limits. We conclude with our vision
and outlook on this subject.
