As written above I had the same issues as most people here. Random reboots, fast draining battery.
I tried reseting the battery, no change.
I checked the battery usage with the app Better Battery Stats and nothing unusual appeared to me. I only realised how often the phone rebooted when it was only lying around.
It seems deleting the Facebook app also did the trick for me. The phone didn't reboot randomly for nearly a week now and battery suddenly lasts for 3 days (instead of 1) like it did about one and a half year ago. Nice!