Electronic devices having housings with image transport layers

An electronic device may have pixels. The pixels may form one or more displays. The displays may be flexible organic light-emitting diode displays or other displays. The electronic device may have first and second display layers that face away from each other and display images in different directions. Image transport layers may overlap the display layers and may have curved edges that overlap a sidewall portion of the electronic device. Image transport layers receive images at input surfaces and transport the received images to corresponding output surfaces. Image transport layers may be provided with hemispherical shapes and other shapes having output surfaces of compound curvature. A folding device may have first and second displays that are overlapped by respective first and second image transport layers that join over a hinge to block the hinge from view. A wristwatch device may have links or other structures with an image transport layer.