These lectures will provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS).

• Basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation.
• Main MPS algorithms, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods.
• Brief introduction to projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction.
• Tensor-network representations of mixed states, quantum channels, and Lindblad dynamics, with applications to thermal states and open quantum systems.

Based on https://arxiv.org/abs/2606.24803.