An open system or open resonator is a domain of wave activity separated from the exterior by a partly open or partly transparent surface. Such open resonators lose energy to infinity through radiation. The numerical computation of the corresponding resonances is complicated by spurious reflections of the outgoing waves at the necessarily finite grid boundaries. These reflections can be reduced to extremely low levels by applying perfectly matched layer (PML) absorbing boundary conditions, which separate the discrete resonances from the continuous spectrum. Using a simple one-dimensional model problem, the influence of the various PML parameters is determined by a numerical error analysis. In addition to one-dimensional open resonators, two-dimensional open resonators as well as various resonating structures in waveguides are considered, and the resonant spectra and selected modes are evaluated. For the first time, leaky modes are computed for several resonating structures in a waveguide in addition to the trapped modes published in the literature. In applications, leaky mode resonances are often more important than trapped mode resonances. Gap tones, observed in a model problem of high-lift configurations, are identified as transversal resonant modes with the lowest radiation losses.