I will provide a brief general introduction into non-Hermitian Hamiltonian systems with real eigenvalue spectra, arguing that they represent well defined self consistent physical systems. Such type of models possess usually an antilinear symmetry, as for instance PT-invariance (simultaneous parity and time reversal) and/or are quasi/pseudo Hermitian. Most crucial is that they allow for a consistent quantum mechanical framework possessing a unitary time evolution. The general framework will be applied to some integrable models, such a quantum spin chains, classical integrable systems associated to differential equations and Calogero-Moser-Sutherland models. I will present some recent results.