The present volume is the second in a two-volume set entitled "Singularities of Differentiable Maps." While the first volume, subtitled "Classification of Critical Points" and originallypublishedas Volume82 in the Monographs in Mathematics series, contained the zoology of differentiable maps, that is, it was devoted to a description of what, where, and how singularities could be encountered, this second volume concentrates on elements of theanatomy and physiology of singularities of differentiable functions. The questions considered are about the structure of singularities and how they function."