Functional renormalization group computation of interacting fermi systems

The functional renormalization group is an ideal tool for dealing with the diversity of energy scales and competition of correlations in interacting Fermi systems. Starting point is an exact hierarchy of flow equations which yields the gradual evolution from a microscopic model Hamiltonian to the effective action as a function of a continuously decreasing energy cutoff. Suitable truncations of the hierarchy have recently led to powerful new approximation schemes. I review applications of the functional renormalization group to the two-dimensional Hubbard model and to one-dimensional Luttinger liquids with impurities.