This expose gives a detailed proof of M. Gromov's pinching theorem for almost flat manifolds. We have two reasons for spending so much effort to rewrite a proof. One is that Gromov's original publication [ 1 ] assumes that the reader is very familiar with several rather different fields and has no difficulties in completing rather unconventional argu-ments - we hope our presentation requires less background. Secondly, we consider the full proof an ideal introduction to qualitative Riemannian geometry since the characteristic interplay between local curvature con-trolled analysis and global geometric constructions occurs at several different levels. These considerations persuaded us to write the following chapters in a selfcontained and hopefully accessible way:
§ 6 treats curvature controlled constructions in Riemannian geometry; § 7 develops metric properties of Lie groups; § 8 explains nonlinear averaging methods; § 2 contains commutator estimates in the fundamental group which are the heart of the Gromov-Margulis discrete group technique and 5.1 is a new form of Malcev's treatment of nilpotent groups.
The proof proper is given in § 3 - § 5, while § 1 contains earlier results and examples pertaining to the almost flat manifolds theorem and a guide to its proof. The statement of the theorem is in 1.5.
We are grateful for discussions with M. Gromov at the I.H.E.S. and the Arbeitstagung in 1977 on the present § 3 after which the idea of this manu-script was born, and at the I.H.E.S. in 1980 which helped to get § 5.1 in its final form. After (countably) many discussions between the two of us we hope that our readers profit from the synthesis of two different styles and temperaments.
Finally our thanks go to Mrs. M. Barron for carefully typing - and retyping -the manuscript and to Arthur L. Besse who suggested contacting Asterisque for publication.