The USPTO unveiled a new patent application by Apple, that describes a mobile device that features a wrap-around flexible AMOLED display (yes, the specifically say it's an OLED display). The whole device is wrapped with the display. The patent also describes a mechanism that determines which parts of the display to use using a camera and a facial recognition software. An OLED is obviously the best choice for a full-color, flexible display that can be efficiently turned-off in some areas.
Apple's patent also describes an option to include multiple transparent displays, overlaid on top of one another. I'm not sure I understand the advantages of using several transparent OLEDs one on top of the other (except for the creation of 3D images). The device shown above represents just one of the concept device designs - Apple also describes "regular" rectangle devices and a cylindrical device. According to the patent the phone body may be totally transparent (made entirely of glass, for example).