  Recently, LHCb experiment announced a 3.5$\sigma$ evidence for direct CP
violation in $D^0$ decay by looking at the difference between
$A_{CP}(D^0\rightarrow K^+K^-)$ and $A_{CP}(D^0\rightarrow \pi^+\pi^-)$. This
is the first evidence of CP violation in charm system, which may indicate new
physics beyond the Standard Model. Motivated by this measurement, we review
rare processes in charmonium decay, especially, the weak decay, $C$ or $P$
violated decay and lepton flavor violated decays. In case the new physics
appears in charm sector, these rare decays of charmonium states will provide
opportunity to search for significant contributions from physics beyond the
Standard Model. With huge $J/\psi$ and $\psi(2S)$ samples in BESIII experiment,
the rare decays may be feasible.
