  Astrophysical observations are a powerful tool to constrain effects of
Lorentz-invariance violation in the photon sector. Objects at high redshifts
provide the longest possible baselines, and gamma-ray telescopes allow us to
observe some of the highest energy photons. Observations include polarization
measurements and time-of-flight measurements of transient or variable objects
to constrain vacuum birefringence and dispersion. Observing multiple sources
covering the entire sky allows the extraction of constraints on anisotropy. In
this paper, I review methods and recent results on Lorentz- and CPT-invariance
violation constraints derived from astrophysical polarization measurements in
the framework of the Standard-Model Extension.
