The family of atomically thin magnets holds great promise for a number of prospective applications in magneto-optoelectronics, with CrI3 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 CrI3 by means of first-principles calculations. We consider a large set of intrinsic impurities and address the 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 and growth temperatures, and to exhibit a strong tendency to agglomerate. In addition, our calculations indicate that the deviation from the nominal stoichiometry largely impacts the magnetic moments, and the defect-induced lattice distortions can drive local ferromagnetic-to-antiferromagnetic phase transitions. Overall, this work portrays a comprehensive picture of intrinsic point defects in monolayer CrI3 from a theoretical perspective.