  We study the electromagnetic (EM) transitions of the $nS$, $nP$ ($n\leq 3$),
and $nD$ ($n\leq 2$) charmonium states with a constituent quark model. We
obtain a reasonable description of the EM transitions of the well-established
charmonium states $J/\psi$, $\psi(2S)$, $\chi_{cJ}(1P)$, $h_c(1P)$ and
$\psi(3770)$. We find that the M2 transitions give notable corrections to some
E1 dominant processes by interfering with the E1 transitions. Our predictions
of EM decay properties for the higher charmonium states are also presented and
compared with other model predictions. In particular, we discuss the EM decay
properties of some $"XYZ"$ states, such as $X(3823)$, $X(3872)$, $X(3915)$,
$X(3940)$ and $X(4350)$ as conventional charmonium states. Assuming $X(3872)$
as the $\chi_{c1}(2P)$ state, our predicted ratio $\Gamma[X(3872)\to
\psi(2S)\gamma]/\Gamma[X(3872)\to J/\psi \gamma]\simeq 4.0$ is consistent with
BaBar's measurement.
