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The aim of this book is to give necessary and sufficient conditions for aC Too map to be CTo-stable: the aim is achieved in a wide range of dimensions via a detailed study of the geometry and topology of many classes of "generic" singularities. Written by internationally renowned authors, the book describes original research, virtually none of which has appeared before in either articles or in book form. The methods developed will stimulate future progress from their application, especially with regard to singularity theory.