A curved display apparatus providing an air touch input function including: a curved display including a concavely curved monitor to display a three-dimensional image; a three-dimensional filter unit having the same radius of curvature as the curved monitor and disposed in front of the curved monitor; a position input unit disposed in front of the three-dimensional filter unit and configured to detect a three-dimensional position according to a user's non-touch motion in a predetermined sensing region; and an image control unit configured to control a three-dimensional image displayed on the curved monitor by transmitting position information sensed through position input unit to the curved display, wherein air touch input through a user's non-touch motion is performed in a concave region of the curved display.