  We calculate the orbital magnetization of 822 two-dimensional magnetic
materials from the Computational 2D Materials Database (C2DB). For compounds
containing 5$d$ elements we find orbital moments of the order of 0.3-0.5
$\mu_\mathrm{B}$, which points to the necessity of including these in any type
of magnetic modeling and comparison with experiments. It is also shown that the
alignment of orbital moments with respect to the spin largely follows the
predictions from Hund's rule and that deviations may be explained by the
$d$-band splitting originating from the crystal field - for example in the
important case of CrI$_3$. Finally, we show that for certain insulators,
Hubbard corrections may lead to large and fully unquenched orbital moments that
are pinned to the lattice rather than the spin and that these moments can lead
to enormous magnetic anisotropies. Such unquenched ground states are only found
from density functional theory calculations that include both Hubbard
corrections and self-consistent spin-orbit coupling and largely invalidates the
use of the magnetic force theorem for calculating magnetic anisotropies.
