These notes are intended to describe the basic concepts of solving inverse problems in a stable way. Since almost all in- verse problems are ill-posed in its original formulation the discussion of methods to overcome difficulties which result from this fact is the main subject of this book. Over the past fifteen years, the number of publications on inverse problems has grown rapidly. Therefore, these notes can be neither a comprehensive introduction nor a complete mono- graph on the topics considered; it is designed to provide the main ideas and methods. Throughout, we have not striven for the most general statement, but the clearest one which would cover the most situations. The presentation is intended to be accessible to students whose mathematical background includes basic courses in ad- vanced calculus, linear algebra and functional analysis. Each chapter contains bibliographical comments. At the end of Chap- ter 1 references are given which refer to topics which are not studied in this book. I am very grateful to Mrs. B. Brodt for typing and to W. Scondo and u. Schuch for inspecting the manuscript.