What does a cell phone service provider require to carry over my service to a new phone?

What does a cell phone service provider require to carry over my service to a new phone?
My current phone is breaking apart so I am buying a new used phone. Verizon will transfer my new number and current service to the new used phone, but what kind of information do they need? I assume that whoever sells me the phone (off of craigslist or ebay most likely) will also need to give me the serial number of that phone. What other information about the phone do I need to make sure the seller has?

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