  Astrophysical black hole candidates are thought to be the Kerr black holes of
general relativity, but there is not yet direct observational evidence that the
spacetime geometry around these objects is described by the Kerr solution. The
study of the properties of the electromagnetic radiation emitted by gas or
stars orbiting these objects can potentially test the Kerr black hole
hypothesis. In this paper, I review the state of the art of this research
field, describing the possible approaches to test the Kerr metric with current
and future observational facilities and discussing current constraints.
