Tensor networks have emerged within the past two decades as a powerful framework to simulate strongly correlated quantum-many-body systems. In particular, it was realized in 2004 that the two powerful and widely successful methods of the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) shared the same algebraic basis in terms of matrix product states (MPS) which thus served as the starting point for this work. While the NRG is truly based on a renormalization group (RG) ansatz, this is not exactly the case for the DMRG, which by now is considered rather a variational ansatz. Even more so, bringing together these two methods in terms of their shared common algebraic basis has proven a very fruitful and instructive approach. It allowed for a better understanding of the NRG through the quantum information concepts borrowed from DMRG. Strict RG related constraints could be loosened by sidestepping to a variational ansatz. The advent of complete basis sets within the NRG, elegantly formulated in MPS, led us to a clear conceptual simplification and streamlining of dynamical quantities within the NRG. With quantum impurity models the standard realm of NRG, this also has seen an increased interest in DMRG simulations with adapted coarse-graining of macroscopic leads in energy space motivated by the NRG. In view of the extreme recent diversification of one-dimensional MPS into more general tensor networks, finally, we developed a powerful tensor library for arbitrary-rank tensors that can deal with any abelian as well as generic non-abelian symmetries beyond SU(2). Powerful applications within the realm of MPS to cutting edge research in physics are demonstrated, with the application to real two-dimensional physical systems kept as an outlook.