An Mg alloy provided with high strength and high ductility by matching the strength and ductility in tensile deformation and compressive deformation at the same levels is provided. The Mg alloy of the present invention is characterized by having a chemical composition consisting of Y: 0.1 to 1.5 at % and a balance of Mg and unavoidable impurities and having a microstructure with high Y regions with Y concentrations higher than an average Y concentration distributed at nanometer order sizes and intervals. The present invention further provides an Mg alloy characterized by having a chemical composition consisting of Y: more than 0.1 at % and a valance of Mg and unavoidable impurities, having a microstructure with high Y regions with Y concentrations higher than an average Y concentration distributed at nanometer order sizes and intervals and having an average recrystallized grain size within the range satisfying the following formula 1: −0.87c+1.10