  Motivated by the potential of atomically-thin magnets towards tunable
high-frequency magnonics, we detail the spin-wave dispersion of bilayer
CrI$_3$. We demonstrate that the magnonic behavior of the bilayer strongly
depends on its stacking configuration and the interlayer magnetic ordering,
where a topological bandgap opens in the dispersion caused by the
Dzyaloshinskii-Moriya and Kitaev interactions, classifying bilayer CrI$_3$ as a
topological magnon insulator. We further reveal that both size and topology of
the bandgap in a CrI$_3$ bilayer with an antiferromagnetic interlayer ordering
are tunable by an external magnetic field.
