It's hard to completely escape safety issues with lithium-ion batteries, in part due to the nature of the electrolytes that charge and release energy when ions shuttle between electrodes. They usually have to be made of easily combustible chemicals to muster enough power. However, that might not be a problem for much longer. Scientists have crafted a water-based electrolyte that's both considerably safer and manages enough voltage (4V) to be useful.
The key was a high concentration of salt that produces the same protective layer on electrodes (a solid electrolyte interphase) you'd get out of a conventional electrolyte, preserving the electrodes and letting the battery hold more energy.
There's one main problem right now: batteries with these watery electrolytes don't last very long. They hold out for just 70 cycles where many conventional lithium-ion batteries last for several hundred cycles or more. However, the very fact that researchers have overcome both safety and voltage issues is still significant. While this won't guarantee that your devices will be completely immune to battery woes, it could be just a matter of time before you panic a little less when you drop your phone.