  Astrophysical black hole candidates are thought to be the Kerr black holes
predicted by General Relativity, but there is not yet clear evidence that the
geometry of the space-time around these objects is really described by the Kerr
metric. In order to confirm the Kerr black hole hypothesis, we have to observe
strong gravity features and check that they are in agreement with the ones
predicted by General Relativity. In this paper, I study the broad K$\alpha$
iron line, which is often seen in the X-ray spectrum of both stellar-mass and
super-massive black hole candidates and whose shape is supposed to be strongly
affected by the space-time geometry. As found in previous studies in the
literature, there is a strong correlation between the spin parameter and the
deformation parameter; that is, the line emitted around a Kerr black hole with
a certain spin can be very similar to the one coming from the space-time around
a non-Kerr object with a quite different spin. Despite that, the analysis of
the broad K$\alpha$ iron line is potentially more powerful than the
continuum-fitting method, as it can put an interesting bound on possible
deviations from the Kerr geometry independently of the value of the spin
parameter and without additional measurements.
