  In the present paper, first the mathematical basic properties of the
exceptional points are discussed. Then, their role in the description of real
physical quantum systems is considered. Most interesting value is the phase
rigidity of the eigenfunctions which varies between 1 (for distant
non-overlapping states) and 0 (at the exceptional point where the resonance
states completely overlap). This variation allows the system to incorporate
environmentally induced effects. In the very neighborhood of an exceptional
point, the system can be described well by a conventional nonlinear
Schr\"odinger equation. In the regime of overlapping resonances, a dynamical
phase transition takes place to which all states of the system contribute: a
few short-lived resonance states are aligned to the scattering states of the
environment by trapping the other states. The trapped resonance states show
chaotic features. Due to the alignment of a few states with the states of the
environment, observable values may be enhanced. The dynamical phase transition
allows us to understand some experimental results which remained puzzling in
the framework of conventional Hermitian quantum physics. The effects caused by
the exceptional points in physical systems allow us to manipulate them for many
different applications.
