Villa Adriana, the summer residence of Emperior Hadrian (used when he didn't travel across his vast empire and had walls built along its borders to protect it), is located a few kilometres outside of Tivoli proper, and can be easily reached by bus from that town (the stop is actually quite close to Villa d'Este). It is a vast archaeological park (the largest palace ever built by a Roman emperor) with many different buildings in various states of preservation - some look like new, some are in ruins, and some look like they have been under renovations for many, many years (e.g. the Teatro Marittimo). A leisurely walk takes at least 2-3 hours, and in some places, away from the crowds of schoolkids, it is easy to imagine yourself in Hadrian's footsteps. He was fond of Greek and Egyptian architecture, and you can easily recognize these influences in many of the buildings, e.g. in the Canopus pond with its Greek statues. The site definitely needs better maintenance (i.e. more funds), but it still gives a fascinating insight into ancient history.