  After the early suggestion by John Pendry to probe unoccupied bands at
surfaces through the time reversal of the photoemission process, the
inverse-photoemission technique yielded the first conclusive experimental
evidence for the existence of image-potential bound states at metal surfaces
and has led over the last two decades to an active area of research in
condensed-matter and surface physics. Here we describe the current status of
the many-body theory of inelastic lifetimes of these image-potential states and
also the Shockley surface states that exist near the Fermi level in the
projected bulk band gap of simple and noble metals. New calculations of the
self-energy and lifetime of surface states on Au surfaces are presented as
well, by using the $GW\Gamma$ approximation of many-body theory.
