This talk will build on results obtained by the author and collaborators in recent years, concerning the adaptation of mortar methods for domain decomposition to the solution of large sliding contact problems in finite deformation solid mechanics and extending these ideas to other numerical methods concerned with interfaces. Such problems are ubiquitous in engineering mechanics. Examples include classical contact-impact, fracture mechanics, propagation of phase boundaries, fluid-structure interaction, and countless others. In general, this talk will focus on issues of numerical stability and accuracy that must be attended to when such interfaces must be included in a realistic simulation of a physical system. In general, in such applications, the interface is characterized by some sort of discontinuity in the fluxes and primary solution variables, subject to interface constitutive laws and/or constraints. The numerical method often represents these features either by a discontinuity in gridding topology as the interface is crossed, or by an enrichment scheme which builds the discontinuities into the local representation of the relevant fields. We will begin by reviewing the development of mortar-based methods for contact-impact analysis in large deformation inelastic solid mechanics. Building upon this basis, recent results will be reviewed in which relevant notions of interface stabilization are brought to bear on polycrystalline interface mechanics in elasticity and in fluid-structure interaction problems.