This volume shows how the analytic properties in the complex energy plane of the Green's functions of many particle systems account for the physical effects (level, shifts, damping, instabilities) characteristic of interacting systems. It concentrates on general physical principles and, while it does not discuss experiments in detail, includes introductions to topics of current research interest, such as singularities (X-ray, Kondo) associated with transient perturbations in an electron gas, the Mott metal - insulator transition in correlated electron systems, and the phenomenon of high T0 superconductivity. This work grew out of a course of graduate lectures given by S. Doniach at the University of London. It will appeal to beginning graduate students in theoretical solid state physics as an introduction to more comprehensive or more specialized texts and also to experimentalists who would like a quick view of the subject. A basic knowledge of solid state physics and quantum mechanics at graduate level is assumed.