Many years have passed since the ?rst edition. However, the encouragements of various readers and friends have persuaded us to write this third edition. During these years, Riemannian Geometry has undergone many dramatic - velopments. Here is not the place to relate them. The reader can consult for instance the recent book [Br5]. of our “mentor” Marcel Berger. However,R- mannian Geometry is not only a fascinating ?eld in itself. It has proved to be a precious tool in other parts of mathematics. In this respect, we can quote the major breakthroughs in four-dimensional topology which occurred in the eighties and the nineties of the last century (see for instance [L2]). These have been followed, quite recently, by a possibly successful approach to the Poincar´ e conjecture. In another direction, Geometric Group Theory, a very active ?eld nowadays (cf. [Gr6]), borrows many ideas from Riemannian or metric geometry. Butletusstophoggingthelimelight.Thisisjustatextbook.Wehopethatour point of view of working intrinsically with manifolds as early as possible, and testingeverynewnotiononaseriesofrecurrentexamples(seetheintroduction to the ?rst edition for a detailed description), can be useful both to beginners and to mathematicians from other ?elds, wanting to acquire some feeling for the subject.