We prove various results about sharply n-transitive sets of homeomorphisms of ``nice'' topological spaces like the real line and the circle. Our main results concern sharply 3-transitive sets G of homeomorphisms of the circle to itself. If G=G &1 , then G contains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharply n-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharply n-transitive sets of homeomorphisms have counterparts in the finite case. 1998 Academic Press, Inc. 1. INTRODUCTION 1.1. Sharply n-Transitive Sets, Affine Planes, Projective Planes, and Minkowski Planes A set G of permutations acting on the set S is called sharply n-transitive, |S| >n, if and only if it acts sharply transitively on the set of all ordered n-tuples of distinct elements of S. A sharply n-transitive set G of permutations of S is invertible if G=G &1 and id S # G. Two sharply n-transitive sets G and H of permutations acting on the set S are isomorphic if there are permutations h 1 and h 2 of S such that h 1 Hh 2 =G. Let G be a set of permutations acting on the set S. We define an incidence structure I G =(P, B G ) consisting of a point set P=S_S and a block set B G consisting of all graphs of functions in G, that is, the sets b g =[(x, g(x)) | x # S] where g # G. The horizontals in P are the sets h a = [(x, a) | x # S], a # S. The verticals in P are the sets v a =[(a, y) | y # G],