The body of mathematics developed in the last forty years or so which can be put under the heading Singularity Theory is quite large. And the excellent introductions to this vast subject which are already available (for instance [AGVJ, [BG], [Gi], [GG], CLm], [Mr], [Ws3 or the more advanced [Ln]) cover necessarily only a part of even the most basic topics.
The aim of the present book is to introduce the reader to a few Important topics from looal Singularity Theory. Some of these topics have already been treated in other introductory books (e.g. right and contact finite determinacy of function germs) while others have been considered only in papers (e.g. Mather's Lemma, classification of simple O-dimensional complete intersection singularities, singularities of hyperplane sections and of dual mappings of projective hypersurfaces!. Even in the first case, we feel that our treatment is different from the introductions mentioned above - the general reason being that we give special attention to the complex analytic situation and to the connections with Algebraic Geometry.