Classical models for fracture and contact with sharp edged rigid punches formulated in the setting of elasticity theory, both nonlinear and linearized, typically predict singular stress and strain fields at crack fronts and punch edges. More generally, singularities arise in elastic boundary value problems for bodies with boundaries containing sharp, re-entrant corners. One perspective holds that the occurrence of singularities in such boundary value problems reflects the omission of certain key physical processes in the associated mathematical models. For the problems cited above, the singularities occur at sharp corners along interfaces separating distinct material phases. Classical models for such problems omit consideration of the complicated physical structure of the interfacial region that separates distinct material phases in real material. This talk discusses a variety of mathematical models for incorporating interfacial mechanics into contact and fracture boundary value problems via non-classical boundary conditions and the impact this has upon the nature of the resulting stress and strain singularities.