Tests of the Lorentz and CPT Symmetries at the Planck Energy Scale with X-Ray and Gamma-Ray Observations

X-ray and gamma-ray observations of astrophysical objects at cosmological distances can be used to probe the energy dependence of the speed of light with high accuracy and to search for violations of Lorentz invariance and CPT symmetry at the Planck energy scale. In this conference contribution, we discuss these searches in the theoretical framework of the Standard-Model Extension. We present new limits on the dispersion relation governed by operators of mass dimension d=5 and d=6, and we discuss avenues for future progress.