The density dependence of the symmetry energy around saturation density, characterized by the slope parameter L, is studied using information provided by the neutron skin thickness in finite nuclei. An estimate for L is obtained from experimental data on neutron skins extracted from antiprotonic atoms. We also discuss the ability of parity-violating elastic electron scattering to obtain information on the neutron skin thickness in 208Pb and to constrain the density dependence of the nuclear symmetry energy. The size and shape of the neutron density distribution of 208Pb predicted by mean-field models is briefly addressed. We conclude with a comparative overview of the L values predicted by several existing determinations.