One of my favorite aspects of wandering though Europe is noticing the layers of civilization--one era folded carefully (or not) into another. For the most part, the U.S. hasn't had to fold back in on itself yet, so our re-purposing of old structures is less noticeable.
For example, the Musee de Cluny. In the 3rd century, it was a Roman Bath. Then in the Middle Ages, an abbey was built on top of the foundation of the baths. Along the way to becoming a public museum, the building was a religious college, a private home, and a "dissection room". Today, those overlapping purposes are exposed to create a home for Medieval art, including famous unicorn tapestries (below).
The Tomb of Napoleon was constructed in a prominent location in the grand Dome Church within the Invalides.
My favorite museum in Paris is probably the Musee d'Orsay, which is a breathtaking transformation of a beautiful train station built for the 1900 world exposition. Prior to becoming a museum in the late 1970's, the building was a WWII package center, a movie set, and a hotel.
On the opposite side of the Seine from the Musee d'Orsay is the recently remodeled Musee de l'Orangerie, which houses several of Monet's famous Nympheas paintings. The building was formerly a greenhouse for orange trees.
Finally, one of the coolest blendings of old and new is the brand new Musee du quai Branly, in the shadow of the Eiffel Tower. As you can see in the photo below, the museum has a classic Parisian frontage which melds into newly-built modern design wings.