Like Hadrian centuries before, Cardinal Ippolito d’Este of Ferrara ordered this villa built on a Tivoli hillside in the mid–16th century. The Renaissance structure, with its second-rate paintings, is not that interesting; the big draw for visitors is the spectacular gardens below (designed by Pirro Ligorio).

As you descend the cypress-studded garden slope you’re rewarded with everything from lilies to gargoyles spouting water, torrential streams, and waterfalls. The loveliest fountain is the Fontana dell’Ovato, by Ligorio. But nearby is the most spectacular engineering achievement: the Fontana dell’Organo Idraulico (Fountain of the Hydraulic Organ), dazzling with its music and water jets in front of a baroque chapel, with four maidens who look tipsy (the fountain “plays” every 2 hours from 10:30am).

The moss-covered Fontana dei Draghi (Fountain of the Dragons), also by Ligorio, and the so-called Fontana di Vetro (Fountain of Glass), by Bernini, are also worth seeking out, as is the main promenade, lined with 100 spraying fountains. The garden is worth hours of exploration, but it involves a lot of walking, with some steep climbs.