  The first mention of the impedance concept appeared on November 1966 in the
CERN internal report "Longitudinal instability of a coasting beam above
transition, due to the action of lumped discontinuities" by V.G. Vaccaro. This
was the beginning of many studies, which took place over the last five decades,
and today, impedances and wake fields continue to be an important field of
activity, as concerns theory, simulation, bench and beam-based measurements.
Building a reliable impedance or wake field model of a machine is the first
necessary step to be able to evaluate the machine performance limitations,
identify the main contributors in case an impedance reduction is required, and
study the interaction with other mechanisms such as optics nonlinearities,
transverse damper, noise, space charge, electron cloud, beam-beam (in a
collider), etc. Beam collective instabilities, and their mitigation, cover a
wide range of effects in particle accelerators and they have been the subjects
of intense research. As the machines performance was pushed new mechanisms were
revealed and nowadays the challenge consists in studying the interplays between
all these intricate phenomena, as it is very often not possible to treat the
different effects separately. With the increasing power of our computers this
becomes easier but the need to continue and develop theories remains, to have a
better understanding of the interplays between all these effects: the subject
of impedance and beam instabilities in particle accelerators is far from being
exhausted, as testified by the many new instability and stabilizing mechanisms
which have been recently explained or discovered. Furthermore, in the context
of the studies for possible future accelerators, some uncharted territories
remain such as, for instance, the collective instabilities during the necessary
ionization cooling for a muon collider.
