In this work, we present some applications of topology optimization for eigenvalue problems where the main goal is to maximize a specified eigenvalue, such as a natural frequency or a linearized buckling load using polygonal finite elements in arbitrary two-dimensional domains. Topology optimization has commonly been used to minimize the compliance of structures subjected to volume constraints. The idea is to distribute a certain amount of material in a given design domain subjected to a set of loads and boundary conditions such that to maximize its stiffness. In this work, the objective is to obtain the optimal material distribution in order to maximize the fundamental natural frequency (e.g. to keep it away from an external excitation frequency) or to maximize the lowest critical buckling load (e.g. to ensure a higher level of stability of the structures). We employ unstructured polygonal meshes constructed using Voronoi tessellations for the solution of the structural topology optimization problems. The design variables, i.e. material densities, used in the optimization scheme, are associated with each polygonal element in the mesh. We present several topology optimization examples for both eigenfrequency and buckling problems in order to demonstrate the functionality and applicability of the proposed methodology.