The Dirichlet-to-Neumann (DtN) map is a nonlocal operator that naturally arises in the study of electrostatics and water waves. Heuristically, it maps the voltage measured at the surface of a conductor to the normal current density through the surface. In this talk, we discuss the sharp two-term spectral asymptotics for the DtN map on 2D curvilinear polygons, determined by the corner angles, edge lengths, and curvature near the corners. This answers three open questions posed in the works of Colbois, Girouard, Gordon, Levitin, Parnovski, Polterovich, Sher, and others. These results are based on a new scattering theory for the DtN map on asymptotically conic domains with Lipschitz boundaries in any dimension. This is joint work with Jeffrey Galkowski and Marcello Malagutti.