An electronic device may have a foveated display, an eye-tracking system and a head movement detection system. The eye-tracking system may gather information on a user's point of regard on the display while the head movement detection system may capture information regarding the rotation of the observer's head. Based on the point-of-regard information, head rotation information, image data, the type of eye/head movement that is underway, and/or tiredness information, control circuitry in the electronic device may produce image data for a display, with areas of different resolutions and(or) visual quality. A full-resolution and(or) quality portion of the image may overlap the point of regard. One or more lower resolution portions of the image may surround the full-resolution portion. The control circuitry may include a gaze prediction system for predicting the movement of the user's gaze during a saccade.