The complex technology of particle accelerators is based upon a series of simple physical concepts. This introduction to the subject focuses on providing a physical understanding of these key ideas. The study surveys the many aspects of accelerator physics and not only explains how accelerators work, but also why the underlying physics leads to a particular choice of design or technique, and points out the limitations of the technology. The mathematical treatment always emphasizes the physical principles described by the equations, and includes a range of calculations which develop a feeling for the quantities and concepts involved.