  The coupling of various charmonium mesons to a photon is studied using
lattice QCD, giving access to radiative form factors and transitions, and
probing the mesons' structure. Methods are developed which allow the robust
determination of amplitudes, including those involving an excited state as well
as multiple form factors, for a range of kinematics. These are applied in
dynamical lattice QCD calculations using the distillation technique to compute
the underlying three-point correlation functions. Form factors and amplitudes
involving the low-lying charmonia, $\eta_c$, $J/\psi$, $\chi_{c0}$ and
$\eta_c'$, are calculated, demonstrating the methods and providing the first
dynamical lattice QCD results for some quantities.
