Researchers from the Korea Advanced Institute of Science and Technology (KAIST) have developed a flexible battery, which is capturing some media attention, and could be used to make electronic devices more bendable.

The important thing to note is that when the battery is bent, it does not interfere with the flow of electricity.

Here it is in action:

A paper from the researchers available (paid) from Nano Letters discusses the battery. The abstract says:

High-performance flexible power sources have gained attention, as they enable the realization of next-generation bendable, implantable, and wearable electronic systems. Although the rechargeable lithium-ion battery (LIB) has been regarded as a strong candidate for a high-performance flexible energy source, compliant electrodes for bendable LIBs are restricted to only a few materials, and their performance has not been sufficient for them to be applied to flexible consumer electronics including rollable displays. In this paper, we present a flexible thin-film LIB developed using the universal transfer approach, which enables the realization of diverse flexible LIBs regardless of electrode chemistry. Moreover, it can form high-temperature (HT) annealed electrodes on polymer substrates for high-performance LIBs. The bendable LIB is then integrated with a flexible light-emitting diode (LED), which makes an all-in-one flexible electronic system. The outstanding battery performance is explored and well supported by finite element analysis (FEA) simulation.

While there are certainly already flexible electronics already in existence, such a battery could remove some of the restrictions of even these, enabling more actual area of the devices to be flexible.