  The observation of ferromagnetic order in a monolayer of CrI$_3$ has been
recently reported, with a Curie temperature of 45 Kelvin and off-plane easy
axis. Here we study the origin of magnetic anisotropy, a necessary ingredient
to have magnetic order in two dimensions, combining two levels of modeling,
density functional calculations and spin model Hamiltonians. We find two
different contributions to the magnetic anisotropy of the material, both
favoring off-plane magnetization and contributing to open a gap in the spin
wave spectrum. First, ferromagnetic super-exchange across the $\simeq $ 90
degree Cr-I-Cr bonds, are anisotropic, due to the spin orbit interaction of the
ligand I atoms. Second, a much smaller contribution that comes from the single
ion anisotropy of the $S=3/2$ Cr atom. Our results permit to establish the XXZ
Hamiltonian, with a very small single ion anisotropy, as the adequate spin
model for this system. Using spin wave theory we estimate the Curie temperature
and we highlight the essential role played by the gap that magnetic anisotropy
induces on the magnon spectrum.
