Active galactic nuclei (AGN) are some of the most energetic sources of radiation in the Universe. While interesting by themselves, their energetic output also profoundly impacts Galaxy evolution. The conversion of gravitational energy into radiation is thought to take place in an accretion disk/corona system just outside the black hole. In this system thermal, UV/optical photons from the accretion disk are upscattered in a corona of hot electrons situated above the accretion disk producing X-rays. The nature of this Comptonizing corona remains a key open question in AGN physics. The recently-launched NuSTAR satellite now provides the opportunity to study the Comptonization spectrum produced by the corona in great detail. In my talk I will show some results from these new studies of the Comptonization spectrum and explore how, together with our growing knowledge of coronal sizes, we are able to draw first conclusions. We find evidence for coronae to be hot and radiatively compact, putting them close to the boundary of the region in the compactness–temperature diagram which is forbidden due to runaway pair production. This suggests that pair production and annihilation are essential ingredients in the coronae of AGN and that they control the coronal temperature and shape of the observed spectra.