  The finite sharply $2$-transitive groups were classified by Zassenhaus in the
1930's. They essentially all look like the group of affine linear
transformations $x\mapsto ax+b$ for some field (or at least near-field) $K$.
However, the question remained open whether the same is true for infinite
sharply $2$-transitive groups. There has been extensive work on the structures
associated to such groups indicating that Zassenhaus' results might extend
might extend to the infinite setting. For many specific classes of groups, like
Lie groups, linear groups, or groups definable in o-minimal it was indeed
proved that all examples inside the given class arise in this way as affine
groups. However, it recently turned out that the reason for the lack of a
general proof was the fact that there are plenty of sharply $2$-transitive
groups which do not arise from fields or near-fields! In fact, it is not too
hard to construct concrete examples (see below). In this note, we survey
general sharply $n$-transitive groups and describe how to construct examples
not arising from fields.
