LG's G Flex phone is now shipping in Korea, and besides the flexible curved OLED display, it also uses a curved battery, developed and produced by LG Chem. The non-removable battery's capacity is 3500mAh. Here's a photo that shows the new battery next to a regular LG mobile phone battery:

LG Chem developed (and patented) a technology which they call Stack & Folding which reduces physical stress when realizing the curved design of battery packs. LG Chem started producing curved batteries this month .

The G Flex has a 6" 720p RGB flexible OLED display made by LG Display that is curved from top to bottom (unlike Samsung's Galaxy Round which is curved from left to right). Other features include a 2.3Ghz Snapdragon S800 CPU, 2GB of RAM, 32GB of internal memory, a 13MP camera, NFC, and Android 4.2.2. The G Flex has two rear-mounted buttons and a "self-healing" back (that will heal scratches quickly).