An electronic device may be provided with wireless circuitry. The wireless circuitry may include one or more antennas. The antennas may include millimeter wave antenna arrays formed from arrays of patch antennas, dipole antennas or other millimeter wave antennas on millimeter wave antenna array substrates. Circuitry such as upconverter and downconverter circuitry may be mounted on the substrates. The upconverter and downconverter may be coupled to wireless communications circuitry such as a baseband processor circuit using an intermediate frequency signal path. The electronic device may have opposing front and rear faces. A display may cover the front face. A rear housing wall may cover the rear face. A metal midplate may be interposed between the display and rear housing wall. Millimeter wave antenna arrays may transmit and receive antenna signals through the rear housing wall.