Nonlocal integral-differential equations and nonlocal balance laws have been proposed as effective continuum models in place of PDEs for a number of anomalous and singular processes. They may also be used to bridge multiscale models, since nonlocality is often a generic feature of model reduction. We discuss a few relevant modeling, computational and analysis issues, including robust simulation codes for validation and verification, effective gradient recovery in a nonlocal setting, and seamless coupling of local (PDEs) and nonlocal models. We demonstrate how new mathematical understanding of nonlocal models and asymptotically compatible discretization can help resolving these issues.