Electrodynamic Properties of a General Physical System -- General Properties of the Linear Optical Response -- Kramers-Kronig Relations and Sum Rules in Linear Optics -- General Properties of the Nonlinear Optical Response -- Kramers-Kronig Relations and Sum Rules in Nonlinear Optics -- Kramers-Kronig Relations and Sum Rules for Harmonic-Generation Processes -- Kramers-Kronig Relations and Sum Rules for Data Analysis: Examples -- Modified Kramers-Kronig Relations in Nonlinear Optics -- The Maximum Entropy Method: Theory and Applications -- Conclusions.This is the first one-volume work to provide a thorough and comprehensive description of the physical background, rigorous theory and applications of Kramers-Kronig relations in the fields of linear and nonlinear optical spectroscopy. Currently, Kramers-Kronig relations have become basic tools in the investigation of the optical properties of materials. A brief presentation of the related data-retrieval technique, the maximum entropy method, is also given. The contents and style potentially make this a standard text for physicists, chemists and engineers interested in optical-materials research and development.