The old town is a UNESCO heritage site, and the ruins of the castle from 963 can still be seen. The Place d'Armes is a large square that is surrounded by lots of restaurants and cafes and is a great place for people watching. This is a relaxing city to tour because there are lots of historical sights to see but most are within easy walking distance of each other. Although the city (and country) are small, it is very clean and there are never the hordes of tourists that you'd encounter in Bruges, Amsterdam or any of the other major Benelux cities.
At night, the lights reflect off the The Cathedral of Notre Dame which contains the tomb of a 14th century Count of Luxembourg. The Petrusse Express is a tourist train that takes people on a tour of the old part of Luxembourg City.
Built between 1572-74 to serve as the City Hall, the Renaissance style building near Place d'Armes has been the Grand Ducal Palace since 1890. It is sometimes open for tours but best to check dates and times at the Tourist Information Centre.