Quantum many-body systems pushed out-of-equilibrium are found to equilibrate to a quasi-stationary state, even if the system is isolated from any environment. The question arises for the nature of the final state and for the time scales on which equilibration takes place. An important aspect is the appearance of a thermal state that should be realised for “non-integrable” systems, which sometimes are seen as the quantum mechanical counterpart to chaotic systems in classical mechanics. Here, we consider the equilibration dynamics of integrable as well as non-integrable 1D systems and describe two interesting findings: 1) for a dimerised system, we can identify equilibration to a (probably) metastable non-thermal state which is called “prethermalization plateau”. 2) in a bosonic system realisable by ultra cold atomic quantum gases, we identify the appearance of stationary states which are well described by negative absolute temperatures and which have been seen in experiment [Braun et al., Science 339, 52 (2013)].