To keep costs down, Motorola shipped the Moto G without a charger; you just get the standard USB cable. It will work with pretty much any standard charger you may have lying around, but some people have reported problems with the phone rebooting when they plug it in, or refusing to charge when the battery runs down. There have also been reports of the Moto G just turning itself off and not charging.
Solution:
This is probably down to the specific charger that you’re using. Try another one and you likely won’t encounter this problem. Ideally, you want a charger rated between 500mA and 1.5A – the higher the rating, the faster it will charge (within these parameters, higher than 1.5A and the Moto G will restrict it anyway).
If you let your Moto G hit zero and it won’t come back to life, try letting it cool down for a while, then plug in a charger (ideally a different one) and leave it for 20 minutes or so before holding down the power key for 20 seconds.