I spent a few months out there in 2004-5. It's a fantastic place, I stayed in Zemun / Novi Beograd (which is not the best place to stay as it's not that near the center). People are all very friendly and if you make an effort to speak a bit of the language (just hello, how are you, etc etc) they will do anything for you.
It's perfectly safe, there are no problems at all, just watch for pickpockets and bag snatchers at some of the really touristy places (e.g. Kalemegdan up on the hill, Sava Church) but I never saw any problems and it's probably safer than places like Charles Bridge in Prague.
Also other cities in Serbia worth visiting are Novi Sad and Kragujevac... both can be done (at a push) in a day from Belgrade. Also you can probably get to lake Ohrid (Macedonia) too in a day... just.
But the city itself is well worth visiting, if you go now the weather is fantastic too (it can be freezing in winter, like -10 and lots of snow) - but in the summer it's warm, on a par with Florence or Rome.
If you have any more questions on it - e.g. good night life, food, getting tickets for matches (Red Star, OFK, Partizan, Serbian National Team) let me know.
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