The field of non-Hermitian physics and especially its degeneracy points, so-called exceptional points, have been receiving increasing interest in various disciplines—from optics, photonics to quantum mechanics or acoustics—for some time now. The non-Hermiticity arises either from an incomplete description of all possible degrees of freedom (open system) or from an amplification or attenuation of the oscillations of a system. The peculiarity of these exceptional points is that for this type of degeneracy not only the eigenvalues of an operator but also its eigenvectors coincide. In addition, these exceptional points are also characterized by an unusual topological structure of the eigenvalues and eigenvectors. The interplay of these properties leads to a plethora of remarkable effects that often defy physical intuition, such as unidirectional invisibility, coherent perfect absorption, increased sensitivity of sensors, or the reversal of the pump dependence of a laser. In addition to these static effects in the vicinity of a non-Hermitian degeneracy, there are also dynamic effects that occur when these points are parametrically orbited. The latter arise not only due to the particular topology, but also, and more importantly, from the fact that the adiabatic theorem is violated in non-Hermitian systems: non-adiabatic transitions between energy levels occur even for very slow parameter variations. This can lead, for example, to an asymmetric state switch when an exceptional point is encircled, where the final state after a cycle depends only on the direction of propagation and not on the initial state. A remarkable effect occurring in the vicinity of exceptional points is utilized to increase the sensitivity of sensors. Here, two degenerate eigenstates are exposed to a small perturbation (signal) causing a frequency splitting. While in the case of conventional Hermitian degeneracies, so-called diabolic points, the frequency splitting correlates only linearly with the strength of the perturbation, in the case of a non-Hermitian singularity the splitting follows a root behavior. Thus, for very small perturbations, the splitting can be orders of magnitude larger compared to the Hermitian case. In this work, we achieve this magnified frequency splitting in a commercially available gyroscope by retrofitting it in such a way that an exceptional point can be reached at steady state. For this purpose, we introduce direction-dependent losses based on the polarization states of light, which we can continuously tune with the help of a half-wave plate. If the gyroscope is now rotated when positioned close to the degeneracy, we observe the characteristic root splitting, which leads to an improved angular velocity measurement. This experiment is not only the first of its kind, it also describes in a convincing way how existing sensors can be adapted to operate close to an exceptional point. The unusual topology of the eigenvalues around a non-Hermitian singularity attracts a significant amount of attention as well. Mathematically, this originates from the fact that the eigenvalues and eigenfunctions in the vicinity of an exceptional point often involve non-holomorphic functions. From a physics perspective, this entails that closed loops around these degeneracies can lead to a swapping of the eigenvalues/eigenvectors. While the associated asymmetric state transfer has already been experimentally confirmed in many areas of physics, new insights and perspectives continuously emerge. In this work, we elucidate the connection between a common protocol for generating a symmetric state transfer called rapid adiabatic passage, which is closely related to Hermitian degeneracies, and the asymmetric state transfer near non-Hermitian degeneracies. For this purpose, we add losses to a conventional method for rapid adiabatic passage such that we create a protocol for encircling an exceptional point, which in turn leads to an asymmetric state exchange. The connection of these previously separate effects is also compellingly confirmed in a microwave waveguide experiment. Finally, this work is also concerned directly with the topology of exceptional points. Although their topology has been extensively studied, theoretical and experimental efforts dealing with dynamical orbits of a non-Hermitian degeneracy have so far been topological only in isolated aspects. The arrangements used so far were based on the concept that the individual eigenstates experience differential linear loss/gain. Moreover, those were exclusively systems in which an injected initial state is brought to a particular final state while executing a single orbit around the degeneracy. It can be shown that in such systems an adiabatic state transfer cannot be achieved for all initial states, and it also generally depends on the encircling direction. However, when the concept of circling around an exceptional point is embedded in a laser cavity, this limitation can be cleverly circumvented such that a topological asymmetric state transfer (besides being fully adiabatic) is generated for both cycling directions. Based on this idea, we introduce a laser in which the light orbits an exceptional point as it passes through two coupled waveguides, thereby emitting two biorthogonal transverse modes simultaneously—one mode at each end of the resonator. The standing wave in the resonator forms a topological lasing mode that mediates an adiabatic transition between the biorthogonal modes at each facet.