  Non-linear effects in accelerator physics are important for both successful
operation of accelerators and during the design stage. Since both of these
aspects are closely related, they will be treated together in this overview.
Some of the most important aspects are well described by methods established in
other areas of physics and mathematics. The treatment will be focused on the
problems in accelerators used for particle physics experiments. Although the
main emphasis will be on accelerator physics issues, some of the aspects of
more general interest will be discussed. In particular, we demonstrate that in
recent years a framework has been built to handle the complex problems in a
consistent form, technically superior and conceptually simpler than the
traditional techniques. The need to understand the stability of particle beams
has substantially contributed to the development of new techniques and is an
important source of examples which can be verified experimentally.
Unfortunately, the documentation of these developments is often poor or even
unpublished, in many cases only available as lectures or conference
proceedings.
