Testing the Kerr black hole hypothesis with RELXILL_NK

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. (paper)