Finite groups equipped with a sharply 2-transitive action have a useful, and simple, geometry associated to them. The geometry allows us to write the finite group as a semi-direct product and recover the structure of a near-field. In this talk, we examine the case of infinite groups with sharply 2-transitive actions. We will use the involutions to study the group, demonstrate how to construct a near field or near domain, and sketch a recent construction of an infinite sharply 2-transitive group that does not split.