An electronic device may be provided with an organic light-emitting diode display with minimized border regions. The display includes a first polymer layer 48, a layer of organic emissive material 46 on the first polymer layer, an encapsulation layer 50 formed over the layer of organic emissive material and a second polymer layer 52 attached to the first polymer layer. The border regions may be minimized by providing conductive structures (microvias 73) that pass through polymer layers 48, 52 of the display and couple conductive traces 51 on the display to conductive traces 59 on additional circuitry 62 that is mounted behind the display.