Let G be a finite group. A proper subgroup H of G is said to be large if the order of H satisfies the bound |H| 3 > |G|. The problem of determining the large maximal subgroups of finite simple groups has a long history, with many applications. In this talk, we give a survey on the study of large maximal subgroups of (almost) simple groups and its applica- tions to group factorisations, combinatorics and geometry. These subgroups arose from the study of triple factorisations G = ABA with A and B non-trivial subgroups of G. Large maximal subgroups also appear in a classification of finite antiflag-transitive generalized quadrangles as well as in the study of flag-transitive and point-primitive designs.
