As a consequence of the classification of the finite simple groups, it hasbeen possible in recent years to characterize Steiner t-designs, that is t -(v, k, 1) designs,mainly for t = 2, admitting groups of automorphisms with sufficiently strongsymmetry properties. However, despite the finite simple group classification, forSteiner t-designs with t > 2 most of these characterizations have remained longstandingchallenging problems. Especially, the determination of all flag-transitiveSteiner t-designs with 3 ≤ t ≤ 6 is of particular interest and has been open for about40 years (cf. Delandtsheer (Geom. Dedicata 41, p. 147, 1992 and Handbook of IncidenceGeometry, Elsevier Science, Amsterdam, 1995, p. 273), but presumably datingback to 1965).The present paper continues the author's work (see Huber (J. Comb. Theory Ser.A 94, 180-190, 2001; Adv. Geom. 5, 195-221, 2005; J. Algebr. Comb., 2007, toappear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We give acomplete classification of all flag-transitive Steiner 5-designs and prove furthermorethat there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on theclassification of the finite 3-homogeneous permutation groups. Moreover, we surveysome of the most general results on highly symmetric Steiner t-designs.