Authors’ abstract: We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two classes of discretization methods are presented, including standard finite element methods and quadrature-based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight into applications and simulations of nonlocal models.