High performance machines such as those used in the semiconductor industry, robotics or racing engines have lots of fast moving parts. The dynamic properties of these moving parts are crucial to the performance of the machine. Therefore these moving parts have to be carefully designed which is often a very time consuming iterative process. In this thesis a general method to optimize the dynamic properties of a structure utilizing topology optimization is investigated. More specifically, the method will be focused on the optimization of eigenfrequencies whilst achieving specific ratios between eigenfrequencies, as dynamic performance requirements are often linked to such criteria. We refer to this class of topology optimization problems as problems involving constrained eigenfrequencies. A particular case of interest is the desired multiplicity of two or more eigenfrequencies, that is a ratio of 1. Several crucial aspects of the topology optimization of eigenfrequencies are investigated, these are the material interpolation methods, mode tracking techniques, multiplicity problems and obtaining a discrete design. By comparing different material interpolation methods, a clear view on the effects of different methods is obtained, leading to solid arguments for selecting a linear material interpolation method for topology optimization of eigenfrequencies. A simple yet effective method of tracking the eigenmodes during the optimization process combined with a solution for the multiplicity problems is presented and verified to show similar results as a more complex analytical approach. A new method of obtaining a discrete design without applying penalization or modification of the eigenvalue problem has been developed using a modified objective function. This method shows promising results and is a good candidate for replacing the material interpolation penalization method. By combining these results, a general and capable framework for the topology optimization of constrained eigenfrequencies is obtained. Using the presented framework, a practical application of the method is given by the design of a cantilever used in an atomic force microscope. Feasible and well-performing designs have been generated, both from a functional and manufacturing point of view.