Particle accelerators in science and technology: status, future developments and challenges Claudio Pellegrini SLAC National Accelerator Laboratory We survey the applications of particle accelerators in science and technology and give a brief history of their development and present status. We introduce the basic theoretical description and physical properties of particle beams and introduce some of the mathematical challenges in evaluating their dynamical behavior. We consider how long time stability in storage rings and self-organizing, collective effects play an important role in determining their characteristics and limitations.