Across various applications, such as diffusion modeling, image processing, and mechanics, continuum models that incorporate nonlocal effects have seen greatly increased use in recent years. These models are characterized by partial integro-differential equations; that is, equations of integral operators that act on difference quotients of multi-variable functions. In this talk, we will discuss recent contributions to the mathematically rigorous underpinning of such nonlocal models across several different contexts. Such contributions include the properties of solutions to nonlocal equations, the robust nature of their discretizations, rigorous characterizations of long-range and other phenomena captured by the equations, and the consistency of nonlocal models with existing classical models in suitable asymptotic regimes. The contexts include continuum mechanics, semi-supervised learning, fractional PDEs, and coupled local-nonlocal equations.