The authors first study hyperaffine planes coordinatized over planar hyperfields. The main purpose of the paper is to show that if an abstract hyperaffine plane satisfies certain additional conditions 1-11, then there exists a hyperfield such that the coordinate plane over this hyperfield is isomorphic to the given hyperaffine plane. It seems that the extra conditions 1-11 are not necessarily true in every hyperaffine plane over a planar hyperfield. This point is not discussed in the paper. The methods are similar, though more involved, to the standard coordinatization of translation planes.