In this talk we present recent developments for the analysis of nonlocal problems with classically-defined, local boundary conditions. We establish an analogue of the fundamental theorem of calculus for the nonlocal operator that recovers the classical boundary flux, which permits the use of variational techniques. Solutions to these nonlocal problems approximate solutions to classical/local differential equations in a suitable regime of nonlocal-to-local convergence, made possible by qualitative properties established for the nonlocal solutions. In certain circumstances, improved modes and rates of variational convergence can be shown. We conclude by discussing applications of these results to convergence in Laplacian learning, nonlocal-fractional transmission problems, and graphon-based variational problems.