The non-locality of quantum many-body systems can be quantified by entanglement measures. Studying the scaling behavior of such measures, one finds that the entanglement in most states of interest (occurring in nature) is far below the theoretical maximum. Hence, it is possible to describe such systems with a reduced set of effective degrees of freedom. This is exploited in simulation techniques based on so-called tensor network states (MPS, PEPS, or MERA). I will describe how this approach can be employed to simulate systems of all particle statistics in order to study ground states, thermal states, and non-equilibrium phenomena. Besides explaining the main ideas, I will highlight some applications. par The second part of the talk focuses on an application to the decoherence in systems that are coupled to an environment. Until our recent study, it was assumed that, as long as the environment is memory-less (i.e. Markovian), the temporal coherence decay is always exponential -- to a degree that this behavior was synonymously associated with decoherence. However, the situation can change if the system itself is a many-body system. For the open spin-1/2 XXZ model, we have discovered that the interplay between dissipation and internal interactions can lead to a divergence of the decohernce time! The quantum coherence then decays according to a power law. To complement the quasi-exact numerical simulation, I will explain the result on the basis of a perturbative treatment.