  When a beam propagates in an accelerator, it interacts with both the external
fields and the self-generated electromagnetic fields. If the latter are strong
enough, the interplay between them and a perturbation in the beam distribution
function can lead to an enhancement of the initial perturbation, resulting in
what we call a beam instability. This unstable motion can be controlled with a
feedback system, if available, or it grows, causing beam degradation and loss.
Beam instabilities in particle accelerators have been studied and analysed in
detail since the late 1950s. The subject owes its relevance to the fact that
the onset of instabilities usually determines the performance of an
accelerator. Understanding and suppressing the underlying sources and
mechanisms is therefore the key to overcoming intensity limitations, thereby
pushing forward the performance reach of a machine.
