This graduate/research level text introduces the theory of multi-electron transitions in atomic, molecular and optical physics, emphasizing the emerging topic of dynamic electron correlation. The book begins with an overview of simple binomial probabilities, classical scattering theory, quantum scattering and correlation, followed by the theory of single electron transition probabilities. Multiple electron transition probabilities are then treated in detail. Various approaches to multiple electron transitions are covered including the independent electron approximation, useful statistical methods and perturbation expansions treating correlation in both weak and strong limits. The dynamics of electron correlation are strongly emphasized. The text contains a comprehensive summary of data for few and many-electron transitions in atoms and molecules, including transitions on different atomic centers, fast ion-atom and electron-atom interactions, and recent observations using synchrotron radiation. Emphasis is given to methods that may be used by nonspecialists.