In the seventies and eighties of the preceding century, existence and classification of vector bundles on projective space $\mathbb P^n$ were all the rage, with contributions from such luminaries as Artin, Atiyah, Hartshorne and Mumford among many others. I have the feeling that not much progress has been made since.

For example, as far as I know, Hartshorne's apparently naïve question "Does there exist an indecomposable algebraic vector bundle of rank 2 on $\mathbb P^n_k \; ?\:"$ is still open for all fields $k$ and all integers $n\geq 6$.

Update[Next day] My colleagues André Hirschowitz and Arnaud Beauville, who are well informed about these questions, have allowed me to report that they feel quite confident that Hartshorne's question is still unsolved.

In the seventies and eighties of the preceding century, existence and classification of vector bundles on projective space $\mathbb P^n$ were all the rage, with contributions from such luminaries as Artin, Atiyah, Hartshorne and Mumford among many others. I have the feeling that not much progress has been made since.

For example, as far as I know, Hartshorne's apparently naïve question "Does there exist an indecomposable algebraic vector bundle of rank 2 on $\mathbb P^n_k \; ?\:"$ is still open for all fields $k$ and all integers $n\geq 6$.