Electronic devices having antennas that radiate through a display

An electronic device may be provided with a display and a phased array antenna that transmits radio-frequency signals at frequencies greater than 10 GHz. The display may include a conductive layer that is used to form pixel circuitry and/or touch sensor electrodes. A filter may be formed from conductive structures within the conductive layer. The conductive structures may include an array of conductive patches separated by slots or may include conductive paths that define an array of slots. The filter may include an additional array of conductive patches stacked under the array of conductive patches to allow the slots to be narrower than would be resolvable to the unaided human eye. The periodicity of the conductive structures and the slots in the filter may be selected to tune a cutoff frequency of the filter to be greater than frequencies handled by the phased antenna array.