I will present a survey on the many-body problem and some numerical techniques to deal with it. In particular I will concentrate on the density matrix renormalization group algorithm, describing its capabilities, limitations and recent breakthroughs inspired by Quantum Information theory.
U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (NY) 326, 96 (2011).
U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).