  The realization of Kitaev's honeycomb magnetic model in real materials has
become one of the most pursued topics in condensed matter physics and materials
science. If found, it is expected to host exotic quantum phases of matter and
offers potential realizations of fault$-$tolerant quantum computations. Over
the past years, much effort was made on 4d$-$ or 5d$-$ heavy transition metal
compounds because of their intrinsic strong spin$-$orbit coupling. But more
recently, there have been growing shreds of evidence that the Kitaev model
could also be realized in 3d$-$transition metal systems with much weaker
spin$-$orbit coupling. This review intends to serve as a guide to this
fast$-$developing field focusing on systems with d$^7$ transition metal
occupation. It overviews the current theoretical and experimental progress on
realizing the Kitaev model in those systems. We examine the recent experimental
observations of candidate materials with Co$^{2+}$ ions: e.g., CoPS$_3$,
Na$_3$Co$_2$SbO$_6$, and Na$_2$Co$_2$TeO$_6$, followed by a brief review of
theoretical backgrounds. We conclude this article by comparing experimental
observations with density functional theory (DFT) calculations. We stress the
importance of inter$-t_{2g}$ hopping channels and Hund's coupling in the
realization of Kitaev interactions in Co$-$based compounds, which has been
overlooked in previous studies. This review suggests future directions in the
search for Kitaev physics in 3d cobalt compounds and beyond.
