  Partial differential equations (PDEs) are used, with huge success, to model
phenomena arising across all scientific and engineering disciplines. However,
across an equally wide swath, there exist situations in which PDE models fail
to adequately model observed phenomena or are not the best available model for
that purpose. On the other hand, in many situations, nonlocal models that
account for interaction occurring at a distance have been shown to more
faithfully and effectively model observed phenomena that involve possible
singularities and other anomalies. In this article, we consider a generic
nonlocal model, beginning with a short review of its definition, the properties
of its solution, its mathematical analysis, and specific concrete examples. We
then provide extensive discussions about numerical methods, including finite
element, finite difference, and spectral methods, for determining approximate
solutions of the nonlocal models considered. In that discussion, we pay
particular attention to a special class of nonlocal models that are the most
widely studied in the literature, namely those involving fractional
derivatives. The article ends with brief considerations of several modeling and
algorithmic extensions which serve to show the wide applicability of nonlocal
modeling.
