  (abridged) With most physicists and astrophysicists in agreement that black
holes do indeed exist, the focus of astrophysical black hole research has
shifted to the detailed properties of these systems. Nature has provided us
with an extremely useful probe of the region very close to an accreting black
hole - X-ray irradiation of relatively cold material in the vicinity of the
black hole can imprint characteristic features into the X-ray spectra of black
hole systems, most notably the Kalpha fluorescent line of iron. Detailed X-ray
spectroscopy of these features can be used to study Doppler and gravitational
redshifts, thereby providing key information on the location and kinematics of
the cold material. This is a powerful tool that allows us to probe within a few
gravitational radii, or less, of the event horizon. Here, we present a
comprehensive review of relativistic iron line studies for both accreting
stellar mass black holes (i.e., Galactic Black Hole Candidate systems; GBHCs),
and accreting supermassive black holes (i.e., active galactic nuclei; AGN).
After a pedagogical introduction to black hole astrophysics, we discuss studies
of relativistic iron lines in the AGN context and show how differences between
classes of AGN can be diagnosed using X-ray spectroscopy. Furthermore, through
a detailed discussion of one particular object (MCG--6-30-15), we illustrate
how the exotic physics of black hole spin are now open to observational study.
We proceed to discuss GBHCs, which turn out to possess rather more complicated
X-ray spectra, making robust conclusions more difficult to draw. However, even
in these cases, modern X-ray observatories are now providing convincing
evidence for relativistic effects. We conclude by discussing the science that
can be addressed by future X-ray observatories.
