  In the recent years, fruitful results on charmed baryons are obtained by
BESIII, Belle and LHCb. We investigate the two-body non-leptonic decays of
charmed baryons in the flavor $SU(3)$ symmetry. Hundreds of amplitude relations
are clearly provided, and are classified according to the $I$-, $U$- and
$V$-spin symmetries. Among them, some amplitude relations are tested by the
experimental data, or used to predict the branching fractions based on the
exact flavor symmetry without any other approximation. Some relations of
$K^0_S-K^0_L$ asymmetries and $CP$ asymmetries are obtained under the $U$-spin
symmetry in the modes of charmed baryon decaying into neutral kaons. Besides,
the $U$-spin breaking effect is explored in the $\Lambda_c^+\to \Sigma^+K^{*0}$
and $\Xi_c^+\to p\bar{K}^{*0}$ modes.
