Can a former U.S. President be Vice-President?

My wife brought up a good question last night: Can a former President be chosen as a VP? I know that they can only be in office for 8 years, or two terms, but can they get into a lower office? Also, if they can, how would it work if s/he was Veep, and the Prez got inated? Would they be allowed past their 8 years? Or would the next in line take office?