Classical, Exceptional, and Exotic Holonomies : a Status Report

I report on the status of the problem of determining the groups that can occur as the irreducible holonomy of a torsion-free affine connection on some manifold. Résumé. Il s’agit d’un rapport sur le problème de la détermination des groupes qui peuvent être les groupes d’holonomie de connexions affines sans torsion. M.S.C. Subject Classification Index (1991) : 53B05 53C10; 53C15 58A15. Acknowledgements. This manuscript was begun during the Round Table at Luminy in 1992. The support of the National Science Foundation in the form of Grant DMS-9205222 is gratefully acknowledged. c © Séminaires & Congrès 1, SMF 1996