How to use Arduino to schedule turning itself off and on

I want my Arduino device to power down when its battery is running low and wait a certain amount of time to reboot after a solar panel has recharged its battery. What can I use to shut it completely down and reboot? I’m looking for something like the Pi Witty real-time-clock and power management board (http://www.uugear.com/product/wittypi2/) except for Arduino. (I’m using the Fuel Gauge product to tell the microcontroller when to shut down).

Thanks for any tips!

One approach is discussed in this tutorial: https://www.gammon.com.au/forum/?id=12821

Thank you–that is a fascinating approach.