The Zn-Mg system has potential glass-forming ability, and therefore studies were made of rapidly solidified zinc-based Zn-Mg alloys containing up to 6 wt % Mg. These alloys exhibited interesting eutectic phase selections and structural transitions across the ribbon thickness which are represented on a microstructure selection diagram for rapid solidification conditions. Although rapid solidification is known in many cases to produce metastable phases, in this case the equilibrium eutectic mixtures of Zn-Mg2Zn11 are observed after rapid solidification, whereas the metastable eutectic mixture Zn-MgZn2 forms under normal solidification conditions. However, in the melt-spun Zn-Mg alloy which is exactly at eutectic composition, th ree different structures are observed across the ribbon thickness. These three structures do not exist simultaneously in the same region, but structural transitions occur as the thickness increases from the wheel side to the free side. Eutectic and hypereutectic alloys show a tendency to form a metallic glass. In these alloys a critical growth velocity exists beyond which eutectic solidification is not possible, suggesting a possible transition from eutectic solidification to amorphous phase formation. The eutectic phase selection and the extent to which a specific microstructure is present depends on the variation in growth rate and solid-liquid interface stability during rapid solidification.