Electrical components such as integrated circuits may be mounted on a printed circuit board. To prevent the electrical components from being subjected to electromagnetic interference, a radio-frequency shielding structure may be mounted over the electrical components. The radio-frequency shielding structure may be formed from a printed circuit that includes a ground plane such as a flex circuit or rigid printed circuit board that includes at least one blanket layer of metal. The printed circuit board to which the electrical components are mounted may include a recess in which the electrical components are mounted. Additional components may be mounted to the interior and exterior surface of the radio-frequency shielding structure. The radio-frequency shielding structure may be formed from a flex circuit that has slits at its corners to accommodate folding.