A curved display device is provided including a display panel having a constant curvature and displaying an image, a backlight unit having the same curvature as that of the display panel, disposed at a rear surface of the display panel, and emitting light to the display panel, and a control unit locally controlling light emission of the backlight unit such that luminance of a light leakage region of the display panel is adjusted. Occurrence of light leakage in the curved display device is prevented, and thus, luminance of the curved display device is kept constant.