This monograph is an attempt to examine and study ridge functions as entities in and of themselves. As such we present what we consider to be various central properties of ridge functions. However, no encyclopedic claims are being made, and the topics chosen are those that we alone considered appropriate. In addition, most chapters contain, either explicitly or implicitly, unresolved questions. It is our hope that this monograph will prove useful and interesting to both researchers and the more casual reader. And, of course, all errors, omissions and other transgressions are totally our responsibility.

No monograph is written in a vacuum, and I would like to especially thank Carl de Boor, Vugar Ismailov and an anonymous referee for various comments and suggestions. Thanks also to Heinz Bauschke, Vitaly Maiorov, Simon Reich and Yuan Xu for their patience, and help with my various inquiries.