Nonlocality is ubiquitous in nature. While partial differential equations (PDE) have been used as effective models of many physical processes, nonlocal balanced laws are also attracting attentions as alternatives to model singularities and anomalous behavior. In this talk, we discuss the mathematical structure of some nonlocal models using a recently developed nonlocal vector calculus that offers an analogy of classical vector calculus for local PDEs. When physically valid, PDEs may be rigorously derived as local limits of their nonlocal counterparts. We also present asymptotically compatible discretizations of nonlocal models that can provide convergent approximations to the local PDE limit. Such discretizations can be more robust for multiscale problems with varying length scales.