This book surveys the physics of the quantum, finite many-body systems that are the basis of nanostructures such as fullerenes and metal clusters. The ab initio techniques for describing the single-particle motion (electrons) and the collective degrees of freedom (plasmons and phonons), and their interaction, are discussed in detail. Applications to the study of phenomena such as the electromagnetic response and superconductivity of these systems are considered. Built around current research and drawing upon lectures given to advanced undergraduates, the book will interest students, young researchers and practitioners in the fields of solid-state and atomic physics and physical chemistry.