This text is an exhaustive account of the modern theory of iterative functional equations. Many of the results included have appeared before only in the research literature. The authors' main purpose is to introduce the reader to the theory and guide him through to the most recent developments and general results. Fundamental notions such as the existence and uniqueness of solutions to the equations are stressed throughout, as are applications of the theory to such areas as branching processes, differential equations, ergodic theory, functional analysis and geometry.