  We discuss the implications of the significant excesses in the diphoton final
state observed by the LHC experiments ATLAS and CMS around a diphoton invariant
mass of 750 GeV. The interpretation of the excess as a spin-zero s-channel
resonance implies model-independent lower bounds on both its branching ratio
and its coupling to photons, which stringently constrain dynamical models. We
consider both the case where the excess is described by a narrow and a broad
resonance. We also obtain model-independent constraints on the allowed
couplings and branching fractions to final states other than diphotons, by
including the interplay with 8 TeV searches. These results can guide attempts
to construct viable dynamical models of the resonance. Turning to specific
models, our findings suggest that the anomaly cannot be accounted for by the
presence of only an additional singlet or doublet spin-zero field and the
Standard Model degrees of freedom; this includes all two-Higgs-doublet models.
Likewise, heavy scalars in the MSSM cannot explain the excess if stability of
the electroweak vacuum is required, at least in a leading-order analysis. If we
assume that the resonance is broad we find that it is challenging to find a
weakly coupled explanation. However, we provide an existence proof in the form
of a model with vectorlike quarks with large electric charge that is
perturbative up to the 100 TeV scale. For the narrow-resonance case a similar
model can be perturbative up to high scales also with smaller charges. We also
find that, in their simplest form, dilaton models cannot explain the size of
the excess. Some implications for flavor physics are briefly discussed.
