I will discuss some general consequences of the LHC diphoton excess as well as implications for concrete models.
For example, the size of the excess imposes a model-independent lower bound on the coupling to photon, which immediately puts strong constraints on possible explanations. Further constraints come from the multitude of run-I resonance searches. The run-I diphoton searches mildly disfavour production from light quarks, but do not rule it out. Searches in final states with leptons, electroweak bosons, and Higgses, provide strong constraints which rule out some models and can and should guide model-building efforts.
Based on these considerations, and utilizing the closeness to the decoupling limit, one can all but rule out interpreting the excess as the H or A in a generic 2HDM. Saving these scenarios seems to require the addition of exotic matter in some form. The same appears to be true for a dilaton interpretation of the excess.
If we require vacuum stability, MSSM candidates (heavy Higgses or sneutrinos) also appear to be ruled out over the whole parameter space; this is true in a leading-order analysis. I briefly comment on higher-order effects and on extensions of the MSSM, as well as on some of the other candidate scenarios that have been proposed so far, such as particles in the SUSY-breaking sector or anomalous pseudo-Nambu Goldstone bosons.