This is an in-depth report on the endotrivial modules, an important class of modular representations for finite groups.Following the historical development of the theory, the book starts with a review of the necessary definitions and some key examples. The main results obtained using traditional techniques are then presented, followed by more recent results such as the work of Grodal inspired by algebraic topology. In the last part of the book original methods are applied to obtain the group of endotrivial modules for certain very important groups.  An accessible reference collecting half a century of research on endotrivial modules, this book will be of interest to researchers in algebra.