The aim of this book is to give necessary and sufficient conditions for a C oo map to be C 0-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. The methods developed for examining the topology and geometry use results from many areas of mathematics - geometric topology, stratification theory, algebraic topology, algebraic geometry, commutative algebra...- and further progress will doubtless be made from the application of deeper results from these areas. Conversely, it is to be hoped that the description of the behaviour of generic singularities will also have interesting consequences for these areas of mathematics, which are those with most interaction with singularity theory. The book describes original research; essentially none of its results has previously appeared elsewhere, either in scientific articles or in books. This book is intended for research mathematicians in singularity theory and in selected areas of geometric topology, stratification theory, algebraic geometry, commutative algebra.