  Quantum spin liquids have fascinated condensed matter physicists for decades
because of their unusual properties such as spin fractionalization and
long-range entanglement. Unlike conventional symmetry breaking the topological
order underlying quantum spin liquids is hard to detect experimentally. Even
theoretical models are scarce for which the ground state is established to be a
quantum spin liquid. The Kitaev honeycomb model and its generalizations to
other tri-coordinated lattices are chief counterexamples --- they are exactly
solvable, harbor a variety of quantum spin liquid phases, and are also relevant
for certain transition metal compounds including the polymorphs of
(Na,Li)$_2$IrO$_3$ Iridates and RuCl$_3$. In this review, we give an overview
of the rich physics of the Kitaev model, including 2D and 3D fractionalization
as well as dynamical correlations and behavior at finite temperatures. We
discuss the different materials, and argue how the Kitaev model physics can be
relevant even though most materials show magnetic ordering at low temperatures.
