  These are lecture notes from the 44th IFF Spring School "Quantum Information
Processing" in Juelich, discussing applications of entanglement theory in
condensed matter. The focus of the notes is on tensor network states, in
particular Matrix Product States (MPS) and Projected Entangled Pair States
(PEPS), which provide an efficient description of many-body states based on
their entanglement structure; both numerical and analytical aspects of tensor
networks are being covered. The notes close with a brief introduction to
quantum complexity, which allows to assess the limitations to our ability to
simulate quantum many-body systems.
