Do the trains in Rome run all night?

I am planning on spending my honeymoon in Italy this fall, we'll be spending a few days in Rome and the traveling down the coast by train, but on the last day we need to stay fairly close to da vinci airport because we have a morning flight, I'm wondering if the trains/subways/busses run all night because I will be staying in the center of Rome the night before my flight (can't afford the airport hotel!) and want to get there on time.