Don't be confused; I'm not talking about paper thin phones like the Nec Media(currently the thinnest phone in the world at 7.7mm, see picture below). I'm talking about the PaperPhone made from the revolutionary computer paper.
Nec Media Phone
"The smartphone prototype, called paperphone is best described as a flexible iphone. The phone does everything a smartphone does; like store books, play music or make phone calls. Its display consists of a 9.5 cm diagonal thin film flexible e ink display. The flexible form of the display makes it much more portable that any current mobile computer: It will shape with your pocket"
Here is how it basically works: the 3.7" flexible electrophoretic (E Ink) display does not consume electricity when it is not refreshed, the little senors on the flim allows the phone to react to user input, and the Android OS lets you play mp3s, read ebooks, make calls, and use various apps.
Keep in mind that this is only the prototype. I am excited where this technology will be able to take the mobile industry.