An electronic device may have a hinge that allows the device to be flexed about a bend axis. A display may span the bend axis. To facilitate bending about the bend axis, the display may have layers such as a display cover layer with grooves or other recesses. The recesses form a flexible portion in the display layer. The display layer may be formed from glass or other materials that are transparent. Elastomeric material, fluids, and other materials may be placed in the recesses in the display layer. The material in the display layer may have an index of refraction that is matched to the index of refraction of the display layer. A hinge may be formed between rigid planar layers that are separated by a gap. Flexible layers that lie flush with opposing surfaces of the rigid planar layers may be used to span the gap.