Tilings of the Plane : From Escher via Möbius to Penrose Part I: Escher seen over his shoulders Part II: Furniture transformations Part III: Penrose tesselation. The aim of the book is to study symmetries and tesselation which have long interested artists and mathematicians. Famous examples are the works created by the Arabs in the Alhambra and the paintings of the Dutch painter Maurits Escher. Mathematicians did not take up the subject intensively until the 19th century. In the process the visualisation of mathematical relationships leads to very appealing images. Three approaches are described in this book. In Part I it is shown that there are 17 principally different possibilities of tesselation of the plane the so called plane crystal groups . Complementary to this ideas of Harald Heesch are described who showed how these theoretical results can be put into practice: He gave a catalogue of 28 procedures that one can use creatively oneself following in the footsteps of Escher so to speak to create artistically sophisticated tesselation. In the corresponding investigations for the complex plane in Part II movements are replaced by bijective holomorphic mappings. This leads into the theory of groups of Möbius transformations: Kleinian groups Schottky groups etc. There are also interesting connections to hyperbolic geometry. Finally in Part III a third aspect of the subject is treated the Penrose tesselation. This concerns results from the seventies when easily describable and provably non periodic parquetisations of the plane were given for the first time. The Contents Part I: Escher seen over the shoulders Part II: Möbius transformations Part III: Penrose tesselation The Author Prof. Dr. Ehrhard Behrends Free University of Berlin Department of Mathematics and Computer Science is the author of numerous mathematical textbooks and popular science books. .