  The accurate characterization of the nuclear symmetry energy and its density
dependence is one of the outstanding open problems in nuclear physics. A
promising nuclear observable in order to constrain the density dependence of
the symmetry energy at saturation is the neutron skin thickness of medium and
heavy nuclei. Recently, a low-energy peak in the isovector dipole response of
neutron-rich nuclei has been discovered that may be correlated with the neutron
skin thickness. The existence of this correlation is currently under debate due
to our limited experimental knowledge on the microscopic structure of such a
peak. We present a detailed analysis of Skyrme Hartree-Fock (HF) plus random
phase approximation (RPA) predictions for the dipole response in several
neutron-rich nuclei and try to elucidate whether models of common use in
nuclear physics confirm or dismiss its possible connection with the neutron
skin thickness. Finally, we briefly present theoretical results for parity
violating electron scattering on ${}^{208}$Pb at the conditions of the PREx
experiment and discuss the implications for the neutron skin thickness of
${}^{208}$Pb and the slope of the symmetry energy.

Nuclear Theory;Solar and Stellar Astrophysics;Nuclear Experiment