This book is the English translation of the French hook "Géométric non commutative" published by Intereditions Paris (1990). The translation wilt be published by Academic Press. After the initial translation of S.K. Berberian, a considerable amount of rewriting and many additions have been done, tripling the size of the original manuscript. In particular the present text contains several unpublished results. My thanks go above all to Cécile whose patience and care for the manuscript have been essential in its completion. This second version of the book greatly benefited of the important modifications suggested by M. Rieflel, D. Sullivan, J.L. Loday, J. Lott, J. Bellissard, P. Cohen, R.. Coiliicrcaux, J. Dixinier, M. Karoubi, P. Kree, H. Bacry, P. de la Harpe, A. Hof.