A curved display apparatus and method for providing a rear combination lamp function are provided. The apparatus includes a laser scanning projector configured to project an image onto a predetermined projection area. A first reflection mirror is configured to reflect the image projected from the laser scanning projector and a second reflection mirror is configured to reflect the image reflected from the first reflection mirror. A curved screen displays the image reflected from the second reflection mirror. A controller is configured to receive data from a data detector that detects a running state, determine an image to be displayed on the curved screen from an image data list set in advance based on the received data, and operate the laser scanning projector based on the determined image.