Partial differential equations are not always suited to model all physical phenomena, especially, if long-range interactions are involved or if the actual solution might not satisfy the regularity requirements associated with the partial differential equation. One remedy to this problem are nonlocal operators, which typically consist of integrals that incorporate interactions between two separated points in space and the corresponding solutions to nonlocal equations have to satisfy less regularity conditions. In PDE-constrained shape optimization the goal is to minimize or maximize an objective functional that is dependent on the shape of a certain domain and on the solution to a partial differential equation, which is usually also influenced by the shape of this domain. Moreover, parameters associated with the nonlocal model are oftentimes domain dependent and thus it is a natural next step to now consider shape optimization problems that are governed by nonlocal equations. Therefore, an interface identification problem constrained by nonlocal equations is thoroughly investigated in this thesis. Here, we focus on rigorously developing the first and second shape derivative of the associated reduced functional. In addition, we study first- and second-order shape optimization algorithms in multiple numerical experiments. Moreover, we also propose Schwarz methods for nonlocal Dirichlet problems as well as regularized nonlocal Neumann problems. Particularly, we investigate the convergence of the multiplicative Schwarz approach and we conduct a number of numerical experiments, which illustrate various aspects of the Schwarz method applied to nonlocal equations. Since applying the finite element method to solve nonlocal problems numerically can be quite costly, Local-to-Nonlocal couplings emerged, which combine the accuracy of nonlocal models on one part of the domain with the fast computation of partial differential equations on the remaining area. Therefore, we also examine the interface identification problem governed by an energy-based Local-to-Nonlocal coupling, which can be numerically computed by making use of the Schwarz method. Here, we again present a formula for the shape derivative of the associated reduced functional and investigate a gradient based shape optimization method.