Fuchsian groups play a central role in various important fields of mathematics. The current book is based on what became known as the famous Fenchel-Nielsen manuscript. Jakob Nielsen (1890-1959) started this project well before World War II, Werner Fenchel (1905-1988) joined later and overtook the much of the preparation of the manuscript. Professor Asmus Schmidt (University of Copenhagen) is the editor of this first publication in book form of the Fenchel-Nielsen notes. It is on his initiative that the long and difficult way of getting the original notes into the proper shape ready for publication succeeded.