  During the past two decades, chiral effective field theory has evolved into a
powerful tool to derive nuclear forces from first principles. Nearly all
two-nucleon interactions have been worked out up to sixth order of chiral
perturbation theory, while, with few exceptions, three-nucleon forces, which
play a subtle, but crucial role in microscopic nuclear structure calculations,
have been derived up to fifth order. We review the current status of these
forces as well as their applications in nuclear many-body systems. While the ab
initio description of light nuclei is generally very successful, we point out
and analyze problems encountered with medium-mass nuclei. We also survey the
construction of equations of state for symmetric nuclear matter and
neutron-rich matter based on chiral forces. A focal point is the symmetry
energy and its impact on neutron skins and systems of astrophysical relevance.
The physics of neutron-rich systems, from nuclei to compact stars, is
essentially determined by the density dependence of the symmetry energy. We
review the status of predictions in comparison with latest empirical
constraints, with particular attention to those extracted from parity violating
electron scattering.
