The Grand Ducal Palace, the official residence of the Grand Duke of Luxembourg, is in Luxembourg City. The palace stands in the historic heart of Luxembourg’s capital, close to the country’s parliament and other central institutions.
The building began life as Luxembourg City’s town hall in 1572. It later served as the seat of a French prefecture and Luxembourg’s government before becoming the governor’s residence in 1817. After Luxembourg became a grand duchy, it developed into the main setting for the monarch’s official duties.
A useful distinction is that the palace is the Grand Duke’s official residence and workplace, but Berg Castle in Colmar-Berg is the family’s principal private residence. The palace hosts meetings, audiences, state banquets and visiting heads of state. Its ceremonial rooms are also opened to visitors for guided tours during the summer months.
Luxembourg City is sometimes confused with Luxembourg, the country, or with Luxembourg Province in Belgium. In this question, the capital-city answer is specifically Luxembourg City.