  Astrophysical black hole candidates are thought to be the Kerr black holes of
general relativity. However, macroscopic deviations from the Kerr background
are predicted by a number of scenarios beyond Einstein's gravity. X-ray
reflection spectroscopy can be a powerful tool to probe the strong gravity
region of these objects and test the Kerr black hole hypothesis. Here I briefly
review the state of the art of this line of research and I present some
constraints on possible deviations from the Kerr metric obtained with the new
X-ray reflection model relxill_nk and XMM-Newton, NuSTAR, and Swift data of the
supermassive black hole in 1H0707-495.
