The rise of nonlocal models as an effective framework for describing a wide range of physical phenomena has spurred intense activity across applied, computational, and theoretical research. While classical continuum mechanics has driven major scientific and technological advances over the past century, many contemporary applications, such as material damage and fracture, phase transitions, and image processing, naturally involve discontinuous, singular, or otherwise low-regularity behavior that lies beyond the reach of purely local theories. This has positioned nonlocal formulations as a natural setting for the study of systems admitting possibly discontinuous solutions. In this talk, I will present several fundamental nonlocal models arising in elasticity, diffusion, and conservation laws, together with geometric notions of curvature tailored to boundaries that lack classical $C^2$ regularity. A central theme will be the role of heterogeneity and boundary interactions, modeled through spatially varying operators and nonlocal boundary conditions. For these systems, I will discuss recent analytical results (including in the nonlinear regime) obtained with students and collaborators. Particular emphasis will be placed on the limiting behavior of heterogeneous nonlocal systems as the interaction horizon tends to zero. In this regime, we identify the corresponding local (differential) models, clarify their physical interpretation, and highlight open problems and future directions motivated by applications in continuum mechanics, biology, image processing, and beyond.