An electronic device may include a display. The display may be an organic light-emitting diode display. The organic light-emitting diode display may have a substrate layer, a layer of organic light-emitting diode structures, and a layer of sealant. Vias may be formed in the substrate layer by laser drilling. The vias may be filled with metal using electroplating or other metal deposition techniques. The vias may be connected to contacts on the rear surface of the display. Components such as flexible printed circuits, integrated circuits, connectors, and other circuitry may be mounted to the contacts on the rear surface of the display.