Polymer adhesion is ubiquitous in both the natural world and human technology. It is also a complex multiscale phenomenon, such that the solution of adhesion problems requires a convergence of chemistry, physics, and engineering. In this Perspective, we provide an overview of some of the fundamental concepts that have emerged in the field of polymer adhesion, discuss recent work, and identify challenges in three specific areas: (a) theories and simulations, with an emphasis on problems involving chain scission; (b) experimental methods for measuring forces and characterizing interfaces at the molecular scale; and (c) strategies inspired by living organisms to generate underwater adhesion.