Simulation-based optimization has become a promising method of designing magnetic confinement devices such as stellarators and particle accelerators. Using simulation optimization (SO) methods to design magnetic confinement systems can be challenging since characteristics of the simulation such as computational noise, stochasticity, and "simulation failures" require special treatment and cannot be handled by out-of-the-box solvers. At the same time, structure in a simulation optimization problem, such as a nonlinear-least-squares objective, can be leveraged to accelerate solution methods for the problem. This dissertation is concerned with understanding the problematic characteristics and leveraging the structure present in simulation optimization problems to efficiently and reliably find promising candidate designs of magnetic confinement devices. We enumerate methods for treating the challenging characteristics, and methods for exploiting structure. To exemplify the many challenges in SO, we solve an array of SO problems in which we seek designs for magnetic confinement devices. The three problems are fundamentally different in nature due to their objective characteristics (smoothness, stochasticity, number of objectives), and formulation of constraints. In the first problem, we seek a Rapid Cycling Synchrotron with integrable optics. We establish feasibility of the highly constrained problem by leveraging the structure present in the constraints (bound, linear, nonlinear simulation-based). We use a structure-aware derivative-free algorithm for solving the non-smooth optimization problem. We perform a sensitivity analysis to understand the dependence of the optimal solution on constraint parameters. Through the second problem, we show how multi-objective optimization (MOO) can be used to understand trade-offs in stellarator design. We discuss the basics of MOO, as well as practical solution methodsfor solving MOO problems. We apply these methods to bring insight into the selection of two common design parameters: the aspect ratio of an ideal magnetohydrodynamic equilibrium, and the total length of the electromagnetic coils. In the third problem, we seek stellarator designs with good fast-ion confinement through the solution of a stochastic optimization problem with a derivative-free algorithm. Methods are compared for computing the stochastic objective. A noise-tolerant optimization method is used to find performant solutions.