LG says it will start churning out flexible OLED displays this year. A company spokesman told The Kora Times that mass production will begin in the fourth quarter at a facility in Paju, South Korea. 12,000 "sheets" are expected to roll off the line each month, though it's unclear how many displays can be created with each one.
The first flexible displays used in consumer products will likely be in the 4-5" range. The Korea Times story claims that LG Electronics, which is a separate entity from LG Display, plans to release a flexible smartphone this year. LG Display expects to sell flexible screens to other device makers, as well.
While flexible displays appear to be in our future, it seems unlikely we'll see smartphones bending in meaningful ways anytime soon. The circuit board, battery, and everything else would have to be similarly pliable. The displays themselves aren't yet flexible enough to be completely rolled up or folded, either.
There is, however, enough flexibility to spread displays over curved surfaces. While such an implementation would seem to have minimal utility for smartphones, it could be ideal for wearable devices like the rumored iWatch. Also, the materials magic that allows flexible displays to bend purportedly makes them unbreakable. An unbreakable smartphone screen certainly has some appeal, even if it's completely flat.