The exterior of the Notre Dame cathedral was undergoing restoration at the time of my first visit (2006). Weather and pollution have taken their toll on this building: large parts of the façade (with the pretty sculptures) are blackened and dirty. The large square in front of the cathedral also is under construction. Maybe the best overall view of the Notre Dame I had from my hotel room window: I could see it bathe in the last streams of the sun in the late afternoon and brightly enlightened at the night.
The interior of the cathedral however is open to visitors every day. It was rather dark inside, because of the mostly cloudy weather. The famous stained glass windows (some relatively recent additions made by Marc Chagall, see 2nd pic) were like stars in the evening sky. There's also a pretty statue of Jeanne d'Arc in the far left corner.
It turned out that I hadn't chosen the best date to visit these French monuments: November 11 is celebrated as a National Holiday (Armistice Day). The Palais du Tau was closed, as was the Abbey of Saint Remi. Both are museums now, and I would have liked a look inside both monumental buildings.
On my second visit, I focused on the Palais du Tau. I had a quick look inside the cathedral first, and again the stained glass windows were what amazed me. The facade is still (or again?) under construction, about half of it looks clean now.
The Palais du Tau lies next to the Cathedral. It now is a museum, holding items related to the cathedral and the coronations. The building itself is pretty modernized, and not too interesting. What I did enjoy were the many sculptures taken from the cathedral: here you can see how huge they are. The museum also holds the original pediment of the cathedral's central portal.
I still did not make it to the Former Abbey of Saint-Remi, which is now a museum with limited opening hours.