Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

Question

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice reference, listing (some of) their invariants? If not, what are the incomplete results?

Edit: I learned that this is a hard open problem. Therefore i would like to restrict the question to the case $n=3$, i.e. "What are the (possibly incomplete) results on classification of vector bundles on $\mathbb{P}_{\mathbb{C}}^3$?"

Thanks!

Answer 1

This doesn't answer your question, but maybe is worth a look. This paper classifies vector bundles on smooth affine threefolds. The methods are highly sophisticated.