In connection to vector bundles over $\mathbb{P}^n$, Hartshorne's paper from 1979 provides a list of open problems. The paper is "Algebraic vector bundles on projective spaces: A problem list" Topology, 18:117–128, 1979.

I don't know which of those problems are still open, but I would be interested in knowing how much progress has been made on those problems, since 1979.