Electronic devices may be provided that contain multiple housing portions. The housing portions may be coupled together using hinges. The hinges may include hinges based on a three-bar linkage, hinges based on a four-bar linkage, hinges with slotted members, hinges formed from flexible support structures, and hinges based on flexible housing structures. Flexible displays may be mounted to the housing portions overlapping the hinges. When the housing portions in a device are rotated relative to each other, the flexible display may bend. The hinge may be configured to allow the flexible display to be placed in a front-to-front configuration in which an active side of the display faces itself or a back-to-back configuration. Engagement structures may be used to help the housing grip external objects and to hold the housing portions together. The hinges may be provided with rotational detents to help hold the flexible display in desired positions.