For many reasons, there exists an ongoing, healthy tension and competition (usually friendly!) between research communities that develop partial-differential equation (PDE) and boundary-integral equation (BIE) models and implementations. In this talk, I will describe two of my recent research efforts employing BIE to model the electrostatic component of biomolecule solvation, emphasizing the connections between PDE and BIE approaches in the hopes of encouraging deeper and more substantive communication and collaboration towards our shared goals: understanding experiments and the underlying mathematics. The first project addresses an electrostatic model I call BIBEE (boundary-integral based electrostatics estimation), which relies on a rigorous operator approximation of the boundary-integral formulation for the common mixed-dielectric Poisson PDE. BIBEE represents the BIE formulation of an earlier approximation by the Borgis group, and here BIE offers novel insights such as the fact that the approximation gives a provable upper bound to the true answer. In the second project, we explore the implications of nonlocal dielectric response by the solvent. This more sophisticated solvation model and its BIE formulation are relatively recent, and a combination of PDE and BIE approaches will undoubtedly offer a much more efficient route to identify and understand the limitations and strengths of this class of solvation models, as well as its connections to other theories of solvation.