Self-adjoint finite-rank perturbations arise naturally, for example, when several boundary conditions of a differential operator are changed simultaneously. The resulting self-adjoint operators’ spectral theory can be described via analytic function theory. This connection has been of interest to the community for some time. Recent advances have shown that (and at times how) their theory is more involved than that of rank one perturbations, beginning with the fact that matrix-valued analytic function theory is required to describe the full picture. We will highlight some key results for finite rank perturbations. Time-permitting, we will discuss applications to some Sturm-Liouville operators and sketch how merging of the theories of boundary triples with finite-rank perturbations provides a more holistic view of an operator’s spectral information.