Testing General Relativity with black hole X-ray data: a progress report

The theory of General Relativity has successfully passed a large number of observational tests. The theory has been extensively tested in the weak-field regime with experiments in the Solar System and observations of binary pulsars. The past 6-7 years have seen significant advancements in the study of the strong-field regime, which can now be tested with gravitational waves, X-ray data, and mm VLBI observations. In my talk, I will summarize the state-of-the-art of the tests of General Relativity with black hole X-ray data, showing the latest results, comparing its constraints on new physics with those from other techniques, and discussing the accuracy of these measurements and the criteria to select sources and observations suitable for these tests.