I present a general introduction to quantum mechanics involving non-Hermitian Hamiltonians. Subsequently I review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. I present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and non-linear integrable field equations of Korteweg-de-Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models we provide three alternative deformations: A complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real valued field equations of non linear integrable systems and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of KdV-type are presented with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanics is presented.