"It is nowadays well established that many-body quantum systems in one and two spatial dimensions exhibit unconventional collective behavior that gives rise to intriguing novel states of matter. Examples are topological states exhibiting nonabelian statistics in 2D and spin-charge separated metals and Mott insulators in 1D. An important focus of current research is to characterize both equilibrium and non-equilibrium dynamics of such systems. The latter has become experimentally accessible only during the last decade and constitutes one of the main frontiers of modern theoretical physics. In recent years it has become clear that entanglement is a useful concept for characterizing different states of matter as well as non-equilibrium time evolution.One main aim of this proposal is to utilize entanglement measures to fully classify states of matter in low dimensional systems. This will be achieved by carrying out a systematic study of the entanglement of several disconnected regions in 1D quantum critical systems. In addition, entanglement measures will be used to benchmark the performance of numerical algorithms based on tensor network states (both in 1D and 2D) and identify the ""optimal"" algorithm for finding the ground state of a given strongly correlated many-body system.The second main aim of this proposal is to utilize the entanglement to identify the most important features of the the non equilibrium time evolution after a ""quantum quench"", with a view to solve exactly the quench dynamics in strongly interacting integrable models. A particular question we will address is which observables ""thermalize"", which is an issue of tremendous current experimental and theoretical interest. By combining analytic and numerical techniques we will then study the non equilibrium dynamics of non integrable models, in order to quantify the effects of integrability."