One of the most important phenomena in magnetism is the exchange interaction between magnetic centers.
In this topical review, we focus on the exchange mechanism in transition-metal compounds and establish kinetic-energy-driven two-sublattice double-exchange as a general exchange mechanism,
in addition to well-known mechanisms like superexchange and double exchange. This mechanism, which was first proposed [Phys. Rev. Lett. {\bf 85}, 2549 (2000)], in the context of Sr$_2$FeMoO$_6$, a
double-perovskite compound, later found to describe a large number of 3d and 4d or 5d transition metal-based double perovskites. The magnetism in
multi-sublattice magnetic systems like double-double and quadrupolar perovskites involving
3d and 4d or 5d transition-metal ions have also been found to be governed by this as a primary mechanism of exchange.
For example, the numerical solution of a two-sublattice double exchange
with additional superexchange couplings for the FeRe-based double double and quadrupolar perovskites are found to reproduce the experimentally observed magnetic ground state as well as the high transition temperature of
above 500 K. The applicability of this general mechanism extends beyond the
perovskite crystal structures, and oxides, as demonstrated for the pyrochlore oxide, Tl$_2$Mn$_2$O$_7$ and the square-net chalcogenides KMnX$_2$ (X=S, Se, Te). The counter-intuitive doping dependence and pressure effect of magnetic transition temperature in Tl$_2$Mn$_2$O$_7$ is explained, while KMnX$_2$ (X=S, Se, Te) compounds are established as half-metallic Chern metals guided by two sublattice double exchange. While the kinetic energy-driven two-site double-exchange mechanism was originally proposed to explain ferromagnetism,
a filling-dependent transition can lead to a rare situation of the antiferromagnetic metallic ground state, as found in La-doped Sr$_2$FeMoO$_6$, and proposed for computer predicted double perovskites Sr(Ca)$_2$FeRhO$_6$. This opens up a vast canvas to explore.