  Functional renormalization group (FRG) methods applied to calculations of
isospin-symmetric and asymmetric nuclear matter as well as neutron matter are
reviewed. The approach is based on a chiral Lagrangian expressed in terms of
nucleon and meson degrees of freedom as appropriate for the hadronic phase of
QCD with spontaneously broken chiral symmetry. Fluctuations beyond mean-field
approximation are treated solving Wetterich's FRG flow equations. Nuclear
thermodynamics and the nuclear liquid-gas phase transition are investigated in
detail, both in symmetric matter and as a function of the proton fraction in
asymmetric matter. The equations of state at zero temperature of symmetric
nuclear matter and pure neutron matter are found to be in good agreement with
advanced ab-initio many-body computations. Contacts with perturbative many-body
approaches (in-medium chiral perturbation theory) are discussed. As an
interesting test case, the density dependence of the pion mass in the medium is
investigated. The question of chiral symmetry restoration in nuclear and
neutron matter is addressed. A stabilization of the phase with spontaneously
broken chiral symmetry is found to persist up to high baryon densities once
fluctuations beyond mean-field are included. Neutron star matter including beta
equilibrium is discussed under the aspect of the constraints imposed by the
existence of two-solar-mass neutron stars.
