Milan's Royal Palace, the Palazzo Reale, stands beside the Duomo and is now one of the city's main temporary exhibition spaces. The building has served Spanish, Austrian, Napoleonic, and Savoy rulers, but most visitors come for the art programme rather than the palace rooms alone.
Exhibitions change often and are ticketed separately, so check what is on before going. Its location makes it easy to combine with the Duomo and Museo del Novecento.