A stellarator confines plasma using non-axisymmetric magnetic fields for fusion energy sciences. This concept provides a promising route to steady-state fusion energy, given its ability to operate without driving a large current in the plasma. Modern stellarator design requires numerical optimization to navigate the high-dimensional spaces used to describe their geometry. Physical insight into the self-adjointness properties of the underlying PDEs -- a nonlinear magneto-fluid equilibrium equation and linear kinetic equation -- enables advanced optimization methods through the efficient calculation of shape derivatives. Adjoint methods, widely applied in the fluid dynamics community, have only recently begun to impact stellarator design. An overview of the optimization approaches and their applications will be presented. Adjoint methods have enabled recent advances in stellarator design, including the efficient calculation of equilibria with record-breaking confinement properties.