  The family of atomically thin magnets holds great promise for a number of
prospective applications in magneto-optoelectronics, with CrI$_3$ arguably
being its most prototypical member. However, the formation of defects in this
system remains unexplored to date. Here, we investigate native point defects in
monolayer CrI$_3$ by means of first-principles calculations. We consider a
large set of intrinsic impurities and address their atomic structure,
thermodynamic stability, diffusion and aggregation tendencies as well as local
magnetic moments. Under thermodynamic equilibrium, the most stable defects are
found to be either Cr or I atomic vacancies along with their complexes,
depending on the chemical potential conditions. These defects are predicted to
be quite mobile at room temperature and to exhibit a strong tendency to
agglomerate. In addition, our calculations indicate that the defect-induced
deviation from the nominal stoichiometry largely impacts the local magnetic
moments, thereby suggesting a marked interplay between magnetism and disorder
in CrI$_3$. Overall, this work portrays a comprehensive picture of intrinsic
point defects in monolayer CrI$_3$ from a theoretical perspective.
