  Radiative transitions between charmonium states offer an insight into the
internal structure of heavy-quark bound states within QCD. We compute, for the
first time within lattice QCD, the transition form-factors of various
multipolarities between the lightest few charmonium states. In addition, we
compute the experimentally unobservable, but physically interesting vector
form-factors of the $\eta_c, J/\psi$ and $\chi_{c0}$.
  To this end we apply an ambitious combination of lattice techniques,
computing three-point functions with heavy domain wall fermions on an
anisotropic lattice within the quenched approximation. With an anisotropy
$\xi=3$ at $a_s \sim 0.1 \mathrm{fm}$ we find a reasonable gross spectrum and a
hyperfine splitting $\sim 90 \mathrm{MeV}$, which compares favourably with
other improved actions.
  In general, after extrapolation of lattice data at non-zero $Q^2$ to the
photopoint, our results agree within errors with all well measured experimental
values. Furthermore, results are compared with the expectations of simple quark
models where we find that many features are in agreement; beyond this we
propose the possibility of constraining such models using our extracted values
of physically unobservable quantities such as the $J/\psi$ quadrupole moment.
  We conclude that our methods are successful and propose to apply them to the
problem of radiative transitions involving hybrid mesons, with the eventual
goal of predicting hybrid meson photoproduction rates at the GlueX experiment.
