  We derive the microscopic spin Hamiltonian for rhombohedral CrI$_3$ using
extensive first-principles density functional theory (DFT) calculations which
incorporate spin-orbit coupling and Hubbard U. Our calculations indicate a
dominant nearest-neighbor ferromagnetic Heisenberg exchange with weaker
further-neighbor Heisenberg terms. In addition, we find a Dzyaloshinskii-Moriya
interaction which primarily drives a topological gap in the spin-wave spectrum
at the Dirac point, and uncover a non-negligible antiferromagnetic Kitaev
coupling between the S=3/2 Cr moments. The out-of-plane magnetic moment is
stabilized by weak symmetric bond-dependent terms and a local single-ion
anisotropy. Using linear spin wave theory, we find that our exchange parameters
are in reasonably good agreement with inelastic neutron scattering (INS)
experiments. Employing classical Monte Carlo simulations, we study the magnetic
phase transition temperature $T_c$ and its evolution with an applied in-plane
magnetic field. We further demonstrate how future high-resolution INS
experiments on the magnon dispersion of single crystals in an in-plane magnetic
field may be used to quantitatively extract the strength of the
antiferromagnetic Kitaev exchange coupling.
