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The past 10–15 years have seen important extensions of the mortar method, a technique for joining dissimilar grids popularized by the domain decomposition community, to the more general problem of contact and impact interactions in finite element analysis. This development has taken place largely in response to several long-standing problems in computational contact mechanics: lack of robustness in solution of the nonlinear and nonsmooth equations of evolution; degradation of spatial convergence rates in problems involving nonconforming meshes on interfaces; lack of a variationally consistent technique for stress recovery on interfaces; and so on. This survey paper summarizes some of the major steps in development of mortar contact formulations. It begins with a basic summary of the mortaring idea in the context of tied contact, it discusses key concepts required for the extension of these methods to large deformation, large sliding formulations of contact-impact, and it previews new results where lessons learned from mortar contact formulations can be extended to a much broader class of interface mechanics applications, considering in particular enriched interface formulations and embedded interface approaches to fluid–structure interaction.