This book is an introductory textbook on the physical processes occurring in the Earth's radiation belts, the van Allen belts. The treatment is quantitative and provides the mathematical basis for original work in this subject. From basic principles the equations are derived describing the motion of energetic ions and electrons trapped in the geomagnetic field. Concepts such as magnetic field representations, guiding center motion, adiabatic invariance, and particle distribution functions are presented in a detailed and accessible manner. Relevant experimental techniques are reviewed and a summary is given of the intensity and energy spectra of the particle populations in the Earth's radiation belts. Problem sets are included as well as appendices of tables, graphs and frequently used formulas.