Belgrade's old fortress (or rather what's left of it) is absolutely worth a visit. It sits on top of a hill, overlooking the the Danube and Sava rivers, New Belgrade, and with a bit of luck the old village of Zemun can be seen in the distance. Take some time walking around the fortifications, or simply sit down someplace to enjoy the view over the city. Inside Kalemegdan's outer walls you'll find the Military Museum, which is definitely worth going to not only if you're a gun nut, it gives an interesting overview of Serbian history. Particularly the WWII section is impressive, although the majority of the info given is available only in Serbian. Just outside the fortress walls you'll find a park, a nice calm spot within the bustling city. Generally, it's quite filled with people, mostly locals, some of them playing chess on tables put there specifically for that purpose.

Old fortress, now, nice park and walking area. Beautiful view to a New and Old Belgrade and the junction of the river Danube and the Sava. One can enjoy visiting good restaurants, ZOO, Military museum or play tennis and basketball.

You can see the ruins and leftovers of the old Belgrade, very nice place for sightseeing, it has two very nice restaurants where you can enjoy the view.

Очень красивое место! Крепость расположена на слиянии рек Сава и Дунай, вечером очень красиво, тихо и спокойно.

It is just beautiful!
I love the view and the fortress walls.