The term Tensor Network (TN) States designates a number of ansatzes that can efficiently represent certain states of quantum many-body systems. In particular, ground states and thermal equilibrium of local Hamiltonians, and, to some extent, real time evolution can be numerically studied with TN methods. Quantum information theory provides tools to understand why they are good ansatzes for physically relevant states, and some of the limitations connected to the simulation algorithms.
 
The potential applications of TNS nowadays extend far beyond quantum many-body physics, for which they were originally introduced. Yet this field still poses some of the most challenging problems for the technique. In particular, in out-of-equilibrium dynamics, the scaling of entanglement with time and system size severely limits a direct application of the methods. However, beyond the standard algorithms, more general TNS techniques can still be used to explore some of the most interesting dynamical properties.