The material used is a commercial magnesium based alloy AZ31(Mg-3Al-1Zn)sheet with a thickness of 1.0mm. Uniaxial tension tests at warm temperature were carried out to investigate the material characteristics of K, m, and n. A warm drawing process with a local heating and cooling technique was developed to improve formability in this study with results of uniaxial tension tests because it is very difficult for Mg alloy to deform at room temperature by the conventional method. The die and blank holder were heated up, while the punch was water-cooled during deformation. FE simulations considering heat transfer were executed with Mg alloy to investigate the Improvement of deep drawability. For the assessment of improvement those were compare with the results of no considering heat transfer and room temperature.