Exceptional points are a special type of degeneracy which can appear for the resonances of parameter-dependent quantum spectra described by non-Hermitian Hamiltonians. They represent positions in the parameter space at which two or even more resonances pass through a branch point singularity. At the critical parameter values, the energies, the widths, and the wave functions describing the resonances are identical. The branching eigenstates show a geometric phase for a parameter space loop

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... the branch point. In this thesis exceptional points are investigated in two important quantum systems. The first system is the hydrogen atom in crossed external electric and magnetic fields. It is a representative for the class of atoms in static external fields, which are as fundamental quantum system accessible both with experimental and theoretical methods and are ideally suited to study the influence of exceptional points. The resonance spectra of the hydrogen atom are numerically calculated with the complex rotation method. A procedure to systematically search for exceptional points is elaborated and the existence of exceptional points is proven. The influence of the branch point singularities on the resonance energies, the wave functions, and the photoionization cross section is analyzed. In addition, a possibility for the observation of exceptional points in an experiment with atoms is proposed. The investigation of the resonances in spectra of the hydrogen atom in this thesis furthermore reveals structures which can provide an insight into the ionization mechanism. The ionization mechanism of the hydrogen atom in crossed electric and magnetic fields has been investigated, e.g., by application of the transition state theory. Here, calculations are performed which give clear evidence for an important influence of the classical transition state in the quantum spectrum. A second class of quantum systems in which exceptional points appear are the stationary states of Bose-Einstein condensates. They are described by the [...]