An electronic device may be provided that has flexible circuitry such as spiral wrapped flexible circuitry. Flexible circuitry may be connected to one or more sides of an electronic component such as rigid printed circuit board or coupled between a rigid printed circuit board an another device component. Flexible circuitry may include an adhesive strip for maintaining a spiral wrap configuration of the flexible circuitry. An adhesive strip may be covered by a removable protective liner during manufacturing of an electronic device so that the flexible circuitry may be tested in a flat, unrolled configuration prior to installation in the electronic device. Flexible circuitry may include a conductive layer configured to form an electromagnetic shield for an electronic component mounted in the spiral wrap. Flexible circuitry may be wrapped around an elongated support member that is mounted along an edge of the electronic component.