  The sensitivities to anomalous quartic photon couplings at the Large Hadron
Collider are estimated using diphoton production via photon fusion. The tagging
of the protons proves to be a very powerful tool to suppress the background and
unprecedented sensitivities down to $6 \cdot 10^{-15}$\gev$^{-4}$ are obtained,
providing a new window on extra dimensions and strongly-interacting composite
states in the multi-TeV range. Generic contributions to quartic photon
couplings from charged and neutral particles with arbitrary spin are also
presented.
