What are the open big problems in algebraic geometry and vector bundles?


More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.



What are some open problems in algebraic geometry?
A few of the more obvious ones:


\* [Resolution of singularities in characteristic p](http://en.wikipedia.org/wiki/Resolution_of_singularities)   

\*[Hodge conjecture](http://en.wikipedia.org/wiki/Hodge_conjecture)  

\* [Standard conjectures on algebraic cycles](http://en.wikipedia.org/wiki/Standard_conjectures) (though these are not so urgent since Deligne proved the Weil conjectures).  

\*Proving finite generation of the canonical ring for general type used to be open though I think it was recently solved; I'm not sure about the details.


For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces. 


(Added later) A very old major problem is that of finding which moduli spaces of curves are unirational. It is classical that the moduli space is unirational for genus at most 10, and I think this has more recently been pushed to genus about 13. Mumford and Harris showed that it is of general type for genus at least 24. As far as I know most of the remaining cases are still open. 


