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This volume consists of selected paper on recent trends and results in the study of various groups of diffeomorphisms, including mapping class groups, from the point of view of algebraic and differential topology, as well as dynamical ones involving foliations and symplectic or contact diffeomorphisms. Most of the authors were invited speakers or participants of the International Symposium on Groups of Diffeomorphisms 2006, which was held at the University of Tokyo (Komaba) in September 2006. This volume is dedicated to Professor Shigeyuki Morita on the occasion of his 60th anniversary. We believe that the scope of this volume well reflects Shigeyuki Morita's mathematical interests. We hope this volume to inspire not only the specialists in these fields but also a wider audience of mathematicians.