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The design of a stellarator with acceptable confinement properties requires optimization of the magnetic field in the non-convex, high-dimensional spaces describing their geometry. Another major challenge facing the stellarator program is the sensitive dependence of confinement properties on electro-magnetic coil shapes, necessitating the construction of the coils under tight tolerances. In this Thesis, we address these challenges with the application of adjoint methods and shape sensitivity analysis. Adjoint methods enable the efficient computation of the gradient of a function that depends on the solution to a system of equations, such as linear or nonlinear PDEs. This enables gradient-based optimization in high-dimensional spaces and efficient sensitivity analysis. We present the first applications of adjoint methods for stellarator shape optimization. The first example we discuss is the optimization of coil shapes based on the generalization of a continuous current potential model. Understanding the sensitivity of coil metrics to perturbations of the winding surface allows us to understand features of configurations that enable simpler coils. We next consider solutions of the drift-kinetic equation. An adjoint drift-kinetic equation is derived based on the self-adjointness property of the Fokker-Planck collision operator, allowing us to compute the sensitivity of neoclassical quantities to perturbations of the magnetic field strength. Finally, we consider functions that depend on solutions of the MHD equilibrium equations. We generalize the self-adjointness property of the MHD force operator to include perturbations of the rotational transform and the currents outside the confinement region. This self-adjointness property is applied to develop an adjoint method for computing the derivatives of such functions with respect to perturbations of coil shapes or the plasma boundary.