Recently, magnesium alloys with a dispersion of quasicrystalline icosahedral i-phase have been shown to exhibit very high strengths combined with ductility. However, effect of the i-phase and its amount has not been investigated comprehensively. To make a systematic investigation, two alloys of composition Mg–6x Zn–x Y, where x=0.5 and 1at%, were chill cast and extruded at three different temperatures to produce various grain sizes with a dispersion of i-phase. The extruded alloys were tested in tension and compression. Very fine grains of micron and submicron size have been obtained, resulting in very high yield strengths of up to about 400MPa accompanied by total elongations of over 12%. Microstructural features such as precipitation and texture have been studied, and mechanical properties such as strength and ductility in tension and compression have been determined. As the grain size is refined, depending on the alloy composition and extrusion temperature, the texture is also modified, such that a higher strength resulting from finer grain size is accompanied by a reasonable elongation, of over 12%. In aged condition the Hall–Petch plots for tensile and compressive yield strengths are nearly parallel, with slopes in the range of 237–307MPaμm−1/2. The Hall–Petch slopes of critically resolved shear stress of slip and twinning are nearly the same at about 63MPaμm−1/2.