As alternatives to partial differential equations (PDEs), nonlocal continuum models given in integral forms avoid the explicit use of derivatives and allow solutions to exhibit desired singular behavior. We present in this talk nonlocal models of mechanics and diffusion processes characterized by a horizon parameter which measures the range of nonlocal interactions. Considering their close connections to classical local PDE models in the limit when the horizon parameter shrinks to zero and to global fractional PDEs in the limit when the horizon parameter tends to infinity, we present numerical schemes that are robust under the changes of the horizon parameter. We also discuss the coupling of models characterized by different scales of horizon.