The Mathieu groups and their geometries Hans Cuypers Eindhoven University of Technology 1. Introduction In the early 80-ties it became clear that the classification of the non-abelian finite simple groups was complete. Among the finite simple groups we find several families: the alternating groups and various families of groups of Lie- type and their twisted analogues. Besides these families there exist 26 sporadic finite simple groups. Buildings are natural geometries for the groups of Lie-type (and, the thin building of type A for the alternating groups). For the 26 sporadic groups there is no canonical type of geometry. The theory of diagram geometries, however, has opened various ways to associate geometries to these sporadic simple groups. In these notes we consider five sporadic simple groups, the Mathieu groups M i , i ∈{11, 12, 22, 23, 24}, as well as some of their geometries. In particular, for the large Mathieu groups, i.e., i ∈{22, 23, 24}, we will describe geometries with the following diagrams: M 22 :    c M 23 :     c M 24 :      c M 22 :    P M 23 :     P M 24 :    L The Mathieu groups were discovered by the French mathematician ´ Emile Mathieu (1835–1890), who also discovered the large Mathieu groups M 22 , M 23 and M 24 . See [12, 13, 14]. They are remarkable groups: for example, apart from the symmetric and alternating groups, M 12 and M 24 are the only 5-transitive 1