Prices are in general on a high level due to the city's status as one of Europe's capitals,
many hotels are to find in the area around the Station.
My recommendation (although quite expensive!) is the Best Western Hotel International (30-32, place de la Gare). Located just opposite the Station and only 5-10 minutes by foot to the old town via Avenue de la Liberté, it is a very good starting point for exploring the city. The rooms are at good, international higher middle-class standard (what a word!), but a bit stereotype; beware: the weekend rate for a single room is 3850 LUF = ca. 95 €!
Very friendly staff and good breakfast, restaurant.
Another tip I've heard of is Hotel Francais which is located right in the heart of the city at Place d'Armes, has an excellent restaurant and is said to offer reasonable prices.
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Address: 20-22 Place De La Gare, Luxembourg City, Luxembourg, 1616, Luxembourg