Given a smooth projective surface, I consider vector bundles of rank $r$ with fixed HarderNarasimhan filtration. Does there exist a moduli space of such bundles? If yes, how is it constructed and which are its properties (dimension, smoothness....)?

Dear ginevra, your question is related to my earlier question Moduli of Extensions. There are some obvious problems which occure already if you fix the HNfactors themselves: Let $E,F$ be sheaves with Ext^1(E,F) onedimensional. Then there are two isomorphism classes of extensions $0 \rightarrow F \rightarrow G \rightarrow E \rightarrow 0$. One is trivial, the other one is not. Moreover the trivial extension is the limit of a isotrivial family of nontrivial extensions. Therefore the modulispace cannot be separated. Nevertheless, you can construct the moduli space of sheaves with HNfactors as an ArtinStack: As pointed out by Arend Bayer, Bridgeland's Introduction to Hallalgebras (arXiv:1002.4372) is a good reference. 

