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Lorentz violation (LV) is predicted by some quantum gravity (QG) candidates,wherein the canonical energy-momentum dispersion relation, $E^2=p^2+m^2$, ismodified. Consequently, new phenomenons beyond the standard model arepredicted. Especially, the presence of LV highly affects the propagation ofastrophysical photons with very high energies from distant galaxies. In thispaper, we review the updating theoretical and experimental results on thistopic. We classify the effects into three categories: (i) time lags betweenphotons with different energies; (ii) a cutoff of photon flux above thethreshold energy of photon decay, $\gamma \rightarrow e^++e^-$; (iii) newpatterns in the spectra of multi-TeV photons and EeV photons, due to theabsorption of background lights. As we can see, the details of LV effects onastrophysical photons depend heavily on the "phase space" of LV parameters.From observational aspects, nowadays available and upcoming instruments canstudy these phenomenons hopefully, and shed light onto LV issues and QGtheories. The most recent progresses and constraints on the ultra-high energycosmic rays (UHECRs) are also fully discussed.

Lorentz violation effects on astrophysical propagation of very high energy photons