Display layers and touch sensor layers may be overlapped by enclosure walls in an electronic device. The electronic device may have a front wall and opposing rear wall and curved sidewalls. The front wall and the curved sidewalls may be formed from a glass layer or other transparent member. A touch sensor layer and display layer may extend under the glass layer with curved sidewalls. A touch sensor layer may also extend under the opposing rear wall. A foldable electronic device may have a flexible transparent wall portion that joins planar transparent walls. Components may be interposed between the transparent planar walls and opaque walls. Display and touch layers may be overlapped by the transparent walls and the transparent flexible wall portion. Touch sensor structures may also be overlapped by the opaque walls.