in the theory of functional differential equations with infinite delay, there are several ways to choose the space of initial functions (phase space); and diverse (duplicated) theories arise, according to the choice of phase space. to unify the theories, an axiomatic approach has been taken since the 1960s. this book is intended as a guide for the axiomatic approach to the theory of equations with infinite delay and a culmination of the results obtained in this way. it can also be used as a textbook for a graduate course. the prerequisite knowledge is foundations of analysis including linear algebra and functional analysis. it is hoped that the book will prepare students for further study of this area, and that will serve as a ready reference to the researchers in applied analysis and engineering sciences.