An electronic device may have a cavity antenna. The cavity antenna may have a logo-shaped dielectric window. An antenna resonating element for the cavity antenna may be formed from conductive traces on a printed circuit board. An antenna resonating element may be formed from the traces. The antenna resonating element may be mounted on an antenna support structure. A conductive cavity structure for the cavity antenna may have a planar lip that is mounted flush with an interior surface of a conductive housing wall. The cavity structure may have more than one depth. Shallower planar portions of the cavity structure may lie in a plane. The antenna resonating element may be located between the plane of the shallow cavity walls and an external surface of the conductive housing wall.