We report a preliminary measurement of the branching fractions of the radiative decays $D^0 \to V \gamma$, where $V=\phi, \bar{K}{}^{*0}$ or $\rho^0$. This is the first observation of the decay $D^0 \to \rho^0 \gamma$. We also present the preliminary results of the first measurement of $CP$ asymmetry in these decays. We present the results of the search for the rare charm decay $D^0 \to \gamma \gamma$, resulting in the most restrictive upper limit to date. Finally, we present the results of the measurement of $CP$ asymmetry in the decay $D^0 \to \pi^0 \pi^0$.