A head-mounted device may have a display for displaying image content. Head-mounted support structures in the device may be used to support the display. The head-mounted device or external equipment may include a three-dimensional image sensor. The three-dimensional image sensor may capture a three-dimensional image of a user's face. Control circuitry may analyze the three-dimensional image to identify which of multiple versions of a customizing face-fit module should be used in the electronic device to fit the device to the user's face. An appropriate face-fit module may be identified based on facial features such as facial size and the shapes of a user's forehead, nose, and cheeks. After identifying which version of the face-fit module is to be used by the user, that version of the face-fit module may be coupled to a non-custom portion of a main unit for the head-mounted device using magnets or other coupling structures.