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

The pygmy dipole strength, the neutron radius of ${}^{208}$Pb and the symmetry energy