The purpose of this study is to confirm the improvement of formability of AZ31 magnesium alloy sheet by using local heating and cooling technique. For this, the experiments of warm deep drawing were done under the temperatures of $100^{circ}C~400^{circ}C$, and the punch velocity of 10, 100mm/min. Also FE analysis under the temperatures of blankholder and die of $150^{circ}C,;225^{circ}C;and;300^{circ}C$ for tools(holder and die) was executed with considering heat teansfer. From the results, the formability of AZ31 magnesium alloy, espicially the temperatures of $225^{circ}C~250^{circ}C$ for tools(holder and die)improved remarkably. And the experiments and simulations showed that necking under room temperature for tools occured under the part of punch shoulder while at $300^{circ}C$ for tools, at the part of die shoulder.