  X-ray studies of stellar mass black holes in X-ray binaries and
mass-accreting supermassive black holes in Active Galactic Nuclei have achieved
a high degree of maturity and have delivered detailed information about the
astrophysical sources and the physics of black hole accretion. In this article,
I review recent progress made towards using the X-ray observations for testing
the "Kerr hypothesis" that the background spacetimes of all astrophysical
quasi-stationary black holes are described by the Kerr metric. Although the
observations have indeed revealed clear evidence for relativistic effects in
strong-field gravity, quantitative tests of the Kerr hypothesis still struggle
with theoretical and practical difficulties. In this article, I describe
several recently introduced test metrics and review the status of constraining
the background spacetimes of mass accreting stellar mass and supermassive black
holes with these test metrics. The main conclusion of the discussion is that
astrophysical uncertainties are large compared to the rather small
observational differences between the Kerr and non-Kerr metrics precluding
quantitative constraints on deviations from the Kerr metric at this point in
time. I conclude with discussing future progress enabled by more detailed
numerical simulations and by future X-ray spectroscopy, timing, polarimetry,
and interferometry missions.
