  Layered transition metal trichalcogenides with the chemical formula $ABX_3$
have attracted recent interest as potential candidates for two-dimensional
magnets. Using first-principles calculations within density functional theory,
we investigate the magnetic ground states of monolayers of Mn- and Cr-based
semiconducting trichalcogenides. We show that the second and third
nearest-neighbor exchange interactions ($J_2$ and $J_3$) between magnetic ions,
which have been largely overlooked in previous theoretical studies, are crucial
in determining the magnetic ground state. Specifically, we find that monolayer
$\text{CrSiTe}_3$ is an antiferromagnet with a zigzag spin texture due to
significant contribution from $J_3$, whereas $\text{CrGeTe}_3$ is a ferromagnet
with a Curie temperature of 106 K. Monolayers of Mn-compounds ($\text{MnPS}_3$
and $\text{MnPSe}_3$) always show antiferromagnetic Neel order. We identify the
physical origin of various exchange interactions, and demonstrate that strain
can be an effective knob for tuning the magnetic properties. Possible magnetic
ordering in the bulk is also discussed. Our study suggests that $\text{ABX}_3$
can be a promising platform to explore 2D magnetic phenomena.
