[Lalaith]

Stockholm:
Skansen and the Vasa museum, boat rides around the islands and the underground food market at Hatorget.

Brussels has some very good bars, or at least it did when I was last there, some years ago now I admit. There was a lovely one shaped just like a 1920s ocean liner (Archiduc), and another which was done up in ultra-goth style, ashtrays like coffins etc. (Le Cercueil)

In Paris, I, like Lush, love Pere Lachaise. Sainte Chapelle is beautiful. Restaurant-wise I rather like Le Procope, which claims to be the oldest restaurant in the city (1686) and has rather a fun and raffish 18th century interior, not hugely expensive either.
As for London....lots of things already said (Gordons is fab) but I would add that despite, or even because of, the opulent naffness, Harrods is always worth a visit - the food hall, the Di and Dodi fountain and the Egyptian staircase are a hoot. And go on one of those open-top bus rides, they are fun and will show you most things you need to see. Walk around the Serpentine in Hyde Park, visit the old Reading Room at the British Museum, and see Hampton Court if you can. Mo Tearoom in Heddon Street, (behind Regent Street, next door to the posh restaurant Momo) is a really good little secret to know about, dead cheap, tasty Moroccan food and very atmospheric.

As an ardent Londoner and European, I have to ask, oblo, what led you to your choice of cities? I've never been to Hamburg....but Mr Lalaith has and he says that other than quite a pretty lake there's not much going on.
Personally, I feel that the most tremendous city in Europe, possibly even more marvellous than London, is Rome.