  We study the semileptonic and non-leptonic charmed baryon decays with $SU(3)$
flavor symmetry, where the charmed baryons can be ${\bf
B}_{c}=(\Xi_c^0,\Xi_c^+,\Lambda_c^+)$, ${\bf
B}'_{c}=(\Sigma_c^{(++,+,0)},\Xi_{c}^{\prime(+,0)},\Omega_c^0)$, ${\bf
B}_{cc}=(\Xi_{cc}^{++},\Xi_{cc}^+,\Omega_{cc}^+)$, or ${\bf
B}_{ccc}=\Omega^{++}_{ccc}$. With ${\bf B}_n^{(\prime)}$ denoted as the baryon
octet (decuplet), we find that the ${\bf B}_{c}\to {\bf B}'_n\ell^+\nu_\ell$
decays are forbidden, while the $\Omega_c^0\to \Omega^-\ell^+\nu_\ell$,
$\Omega_{cc}^+\to\Omega_c^0\ell^+\nu_\ell$, and $\Omega_{ccc}^{++}\to
\Omega_{cc}^+\ell^+\nu_\ell$ decays are the only existing Cabibbo-allowed modes
for ${\bf B}'_{c}\to {\bf B}'_n\ell^+\nu_\ell$, ${\bf B}_{cc}\to {\bf
B}'_c\ell^+\nu_\ell$, and ${\bf B}_{ccc}\to {\bf
B}_{cc}^{(\prime)}\ell^+\nu_\ell$, respectively. We predict the rarely studied
${\bf B}_{c}\to {\bf B}_n^{(\prime)}M$ decays, such as ${\cal
B}(\Xi_c^0\to\Lambda^0\bar K^0,\,\Xi_c^+\to\Xi^0\pi^+)=(8.3\pm 0.9,8.0\pm
4.1)\times 10^{-3}$ and ${\cal B}(\Lambda_c^+\to
\Delta^{++}\pi^-,\,\Xi_c^0\to\Omega^- K^+)=(5.5\pm 1.3,4.8\pm 0.5)\times
10^{-3}$. For the observation, the doubly and triply charmed baryon decays of
$\Omega_{cc}^{+}\to \Xi_c^+\bar K^0$, $\Xi_{cc}^{++}\to (\Xi_c^+\pi^+$,
$\Sigma_c^{++}\bar K^0)$, and $\Omega_{ccc}^{++}\to (\Xi_{cc}^{++}\bar
K^0,\Omega_{cc}^+\pi^+,\Xi_c^+ D^+)$ are the favored Cabibbo-allowed decays,
which are accessible to the BESIII and LHCb experiments.
