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Due to their high complexity, our understanding of strongly correlated quantum many-body systems is still limited. This concerns, for example, both static and dynamical equilibrium properties of fermionic systems and quantum magnets in condensed matter physics, as well as complex molecules with strong entanglement in quantum chemistry. Other big challenges are the analysis of non-equilibrium dynamics of closed systems, systems interacting with an environment, and transport problems. Tensor network states such as MPS, TTN, PEPS, and MERA are designed to capture the structure of entanglement in quantum systems in compressed representations. They are also particularly well-suited for the study of topological ordered phases of matter which cannot be described within the framework of spontaneous symmetry breaking. A better mathematical understanding of approximate tensor network representations is emerging for cycle-free networks like MPS and TTN.