  Luminous accreting stellar mass and supermassive black holes produce
power-law continuum X-ray emission from a compact central corona. Reverberation
time lags occur due to light travel time-delays between changes in the direct
coronal emission and corresponding variations in its reflection from the
accretion flow. Reverberation is detectable using light curves made in
different X-ray energy bands, since the direct and reflected components have
different spectral shapes. Larger, lower frequency, lags are also seen and are
identified with propagation of fluctuations through the accretion flow and
associated corona. We review the evidence for X-ray reverberation in active
galactic nuclei and black hole X-ray binaries, showing how it can be best
measured and how it may be modelled. The timescales and energy-dependence of
the high frequency reverberation lags show that much of the signal is
originating from very close to the black hole in some objects, within a few
gravitational radii of the event horizon. We consider how these signals can be
studied in the future to carry out X-ray reverberation mapping of the regions
closest to black holes.
