In a semiconductor device manufacturing method, a target object including a multilayer film and a mask formed on the multilayer film is prepared in a processing chamber of a plasma processing apparatus. The multilayer film is formed by alternately stacking a silicon oxide film and a silicon nitride film. The multilayer film is etched by supplying a processing gas containing hydrogen gas, hydrogen bromide gas, nitrogen trifluoride gas and at least one of hydrocarbon gas, fluorohydrocarbon gas and fluorocarbon gas into the processing chamber of the plasma processing apparatus and generating a plasma of the processing gas in the processing chamber.