Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on $CP^5$. In particular, I will describe a classification of such bundles which involves a connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on $CP^5$ from ``simple'' ones; and future directions related to this project.