Renormalization and tensor product states in spin chains and lattices

We review different descriptions of many-body quantum systems in terms of tensor product states. We introduce several families of such states in terms of the known renormalization procedures, and show that they naturally arise in that context. We concentrate on matrix product states, tree tensor states, multiscale entanglement renormalization ansatz and projected entangled pair states. We highlight some of their properties, and show how they can be used to describe a variety of systems.