  We present searches for rare charm decays of the form $X_c^+ \to h^{\pm}
\ell^{\mp} \ell^{(')^+}$, where $X_c^+$ is a charm hadron either $D^+$,
$D_s^+$, or $\Lambda_c^+$, and $\ell^{(')^{\pm}}$ is an electron or muon. These
modes are based on 384 $fb^{-1}$ of $e^+ e^-$ annihilation data collected at
the $\Upsilon(4S)$ resonance with the BABAR detector at the SLAC National
Accelerator Laboratory. We also present the flavor-changing neutral-current
decays $D^0 \to e^+e^-$, $D^0 \to \mu^+ \mu^-$, and $D^0 \to e^{\pm} \mu^{\mp}$
that corresponds to an integrated luminosity of 468 $fb^{-1}$ of data. The
decay $D^0 \to e^+ \mu^-$ is further lepton-flavor violating, and thus occur
only through very slow neutrino mixing. These decays constitute sensitive
probes for possible new-physics contribution. We report new limits on the
branching fractions of these decays.
