What you are looking at is a curved edge phone from LG. In fact both sides are curved. Samsung at the end of 2014 released the Note Edge and if rumours are to be believed there will be a Samsung Galaxy S6 with 2 curved edges.
In terms of usability I found the curved edge of my Note Edge fantastic, it just had poor battery life. But if curved edges are to be the new must have feature can you really see yourself buying a phone for this feature ? And would you pay a premium for this design element?