It is notoriously difficult to construct stable bundles with given topological invariants. This difficulty is partially because the relationship between categorical constructions and stability is unclear. For example, it is unknown whether there exist general conditions that guarantee the kernel of a morphism between stable bundles is still stable. Recent work has focused on a specific case of this general problem: stability of syzygy bundles. Syzygy bundles are bundles that arise as the kernel of the evaluation map on global sections. In this talk, we discuss recent progress on this problem.