  A decade ago, Alexei Kitaev proposed an exactly solvable $S$ = 1/2 model on a
two-dimensional honeycomb lattice, where the spins fractionalize into Majorana
fermions and form a topological quantum spin liquid (QSL) in the ground state.
It was soon recognized that a family of complex iridium oxides, as well as
ruthenium chloride, with honeycomb structure are magnetic insulators and
accommodate essential ingredients of the Kitaev model, due to the interplay of
electron correlation and spin-orbit coupling. This initiated a race to
materialize the Kitaev QSL and to capture the signature of Majorana fermions.
In this review, we provide a wide perspective of this rapidly growing field,
including theory, materials and experiment. We first summarize the theoretical
background of the Kitaev QSL ground state and its materialization using
spin-orbital-entangled $J_{\rm eff}$ = 1/2 moments. This is followed by an
overview of candidate materials and their magnetic properties, including
Na$_2$IrO$_3$, $\alpha$, $\beta$, $\gamma$-Li$_2$IrO$_3$,$\alpha$-RuCl$_3$ and
H$_3$LiIr$_2$O$_6$. Finally, we review the latest exciting progress in the
search for the Kitaev QSL. In particular, H$_3$LiIr$_2$O$_6$ and
$\alpha$-RuCl$_3$ in applied magnetic field show signatures of the QSL state,
and $\alpha$-RuCl$_3$ has unusual magnetic excitations and thermal transport
properties that are consistent with spin fractionalization.
