An electronic device may be provided with an antenna module and a phased antenna array on the module. The module may include a logic board, an antenna board surface-mounted to the logic board, and a radio-frequency integrated circuit (RFIC) mounted surface-mounted to the logic board. The phased antenna array may include antennas embedded in the antenna board. The antennas may radiate at centimeter and/or millimeter wave frequencies. The logic board may form a radio-frequency interface between the RFIC and the antennas. Transmission lines in the logic board and the antenna board may include impedance matching segments that help to match the impedance of the RFIC to the impedance of the antennas. The module may efficiently utilize space within the device without sacrificing radio-frequency performance.