Resonances in open systems can be described by eigenvalue problems with a radiation condition at infinity and arise in various fields including acoustics, classical mechanics, quantum mechanics, and x-ray physics. This thesis focusses on the optimization of resonances for multilayer x-ray resonators which consist of several layers and support certain resonant states. These resonant states can be excited by x-ray beams under special grazing angles of incidence (corresponding to resonant frequencies of the system) leading to a very high field enhancement inside the system compared to the incident field. X-ray resonators or waveguides can be used for filtering, guiding, and concentration of x-rays, which is for example useful in nanoscale x-ray structure analysis and x-ray imaging. The multilayer structures can be characterized by the refractive index n. We want to find a function n for which the field enhancement in the multilayer structure for a resonant angle of incidence is maximized subject to side constraints on n. For the optimization problem we use an objective function involving complex resonances and corresponding resonance functions. Analytic expressions for the derivatives of resonances and resonance functions with respect to n are derived using perturbation theory of linear operators. As a side product, approximation formulas for the reflectivity are obtained, in particular a mathematical justification for the widely used kinematic approximation. Higher order Taylor approximations and Pad´e approximations lead to significant improvements of the standard approximation formulas, especially close to the critical angle. We explain how the optimization problem can be discretized and finish with numerical computation leading to improved multilayer x-ray resonators for several situations.