I provide a general introduction into the problematic arising when considering quantum mechanical systems, which are described by non-Hermitian Hamiltonians and discuss how they may be overcome. I present a systematic method to construct non-Hermitian Hamiltonians, which possess a real positive spectrum. Some of the newly constructed Hamiltonians are simple non-local one-particle systems, which are non-Hermitian perturbations of the harmonic or anharmonic oscillator others describe integrable many body systems such as generalised versions of Calogero-Moser-Sutherland models. I discuss a scheme which allows to compute physical quantities when such systems are time dependent. In particular I consider systems driven by time-dependent laser fields and present some results of these explicit computations.