An electronic device may be provided with a display. The display may have a flexible display layer covered by a protective display cover layer. The flexible display layer may be an organic light-emitting diode display layer or other layer with a flexible substrate. The flexible display layer may have a central region with peripheral edges. The central region may be rectangular, may be octagonal, or may have other shapes. Strip-shaped protrusions may extend along each of the peripheral edges of the central region. The strip-shaped protrusions may be bent to provide the protrusions with curved surface profiles. The display cover layer may have a cushion shape with bowed edges that overlap parts of the strip-shaped protrusions, may have curved inner and outer surfaces and/or planar surfaces, may have rounded corners with compound curvature, may have edges with curved surface profiles, and/or may have other configurations.