From the 14th through the 18th centuries, the rulers of Genoa were called doges, and they ruled from the Doge's Palace – the Palazzo Ducale in Italian – in the historic city center. Today, the palace is open to the public as a museum.
The Palazzo Ducale was built starting in the 1250s, although the finishing touches on the building weren't complete until the 1530s. The palace once served as both the residence for the ruling doge and the offices from which he would govern the Republic of Genoa. The palace was added to at various points over the years, and partially rebuilt twice (once after a fire in the 1770s).
There are two main entrances to the Palazzo Ducale. The main entrance is on the Piazza Matteotti, and a secondary entry is on the famous Piazza de Ferrari. Today, the palace serves various civil functions. There are regular exhibitions held in the palace, including visiting contemporary art shows, as well as a couple of large halls that are often used for events. Those halls are decorated with historic frescoes, making the venue part of the highlight of any event held there.
The Palazzo Ducale is open every day, but since the exhibitions change regularly you'll need to consult the tourist information office in Genoa or simply visit the Doge's Palace to find out what's on display when you're in the city.