In this volume of the <b>Encyclopaedia</b>, the authors give a tour of the principal areas and methods of modern differential geometry. Some topics, though important, have not previously appeared in books with a general readership. The book is structured to allow the reader to skip about the text, rather than systematically reading from beginning to end. Compared to other <b>EMS</b> volumes, <b>EMS 28</b> has an increased appeal for early graduate students.