In the summer semester of 1972 I gave a course of lectures on the local theory of differentiable maps at the University of Freiburg. These lectures have formed the basis for the first thirteen chapters of the book, the next three chapters having been written for a summer school organised by the Studienstiftung des deutschen Volkes. My students were responsible for removing many mistakes from the original manuscript which has now been translated into English by L. Lander. He has also made a number of improvements and corrections and provided the last chapter together with its pictures and list of publications. The later chapters discuss a subject which has been the real motivation for writing the book: classical catastrophe theory.

We have both profited greatly from a lecture course on catastrophe theory by K. Jänich, given in Regensburg during the winter semester 1971/72, which contained most of the information and pictures presented in chapter 17.

A small number of copies of the German text of the present book were printed for our students under the title: Der Regensburger Trichter, Band 3, Differenzierbare Abbildungen.

On the pages that follow, the reader will not find any new results or methods. Our purpose is to make it easier for those students, who have properly understood the basic lecture courses on analysis and possess a basic knowledge of algebra, to learn about recent work on differentiable maps, in particular, the mysteries of catastrophe theory.