  Motivated by the recent LHC observation of an excess of diphoton events
around an invariant mass of 750 GeV, we discuss the possibility that this is
due to the decay of a new scalar or pseudoscalar resonance dominantly produced
via photon-photon fusion. We present a precise calculation of the corresponding
photon-photon luminosity in the inclusive and exclusive scenarios, and
demonstrate that the theoretical uncertainties associated with these are small.
In the inclusive channel, we show how simple cuts on the final state may help
to isolate the photon-photon induced cross section from any gluon-gluon or
vector boson fusion induced contribution. In the exclusive case, that is where
both protons remain intact after the collision, we present a precise cross
section evaluation and show how this mode is sensitive to the parity of the
object, as well as potential $CP$-violating effects. We also comment on the
case of heavy-ion collisions and consider the production of new heavy
colourless fermions, which may couple to such a resonance.
