These notes give a fairly elementary introduction to the local theory of differentiable mappings. Sard´s Theorem and the Preparation Theorem of Malgrange and Mather are the basic tools and these are proved first. There follows a number of illustrations including: the local part of Whitney´s Theorem on mappings of the plane into the plane, quadratic differentials, the Instability Theorem of Thom, one of Mather´s theorems on finite determinacy and a glimpse of the theory of Toujeron. The later part of the book develops Mather´s theory of unfoldings of singularities. Its application to Catastrophe theory is explained and the Elementary Catastrophes are illustrated by many pictures. The book is suitable as a text for courses to graduates and advanced undergraduates but may also be of interest to mathematical biologists and economists.
