An X-ray exposure apparatus operating in a vacuum or a reduced-pressure environment includes a mask, arranged in the vacuum or in the reduced-pressure environment, for holding a reflection X-ray mask having a mask pattern thereon, an X-ray illuminating system arranged so as to illuminate the reflection X-ray mask having the mask pattern thereon, relatively scanningly with X-rays, wherein the mask pattern being illuminated is transferred onto an object, and a cooling structure for cooling the X-ray mask held by the mask chuck.