  In this review, we summarize recent studies on nuclear matter and finite
nuclei based on parity doublet models. We first construct a parity doublet
model (PDM), which includes the chiral invariant mass $m_0$ of nucleons
together with the mass generated by the spontaneous chiral symmetry breaking.
We then study the density dependence of the symmetry energy in the PDM, which
shows that the symmetry energy is larger for smaller chiral invariant mass.
Then, we investigate some finite nuclei by applying the Relativistic Continuum
Hartree-Bogoliubov (RCHB) theory to the PDM. We present the root-mean-square
deviation (RMSD) of the binding energies and charge radii, and show that $m_0$
= 700 MeV is preferred by the nuclear properties. Finally, we modify the PDM by
adding the iso-vector scalar meson $a_0(980)$ and show that the inclusion of
the $a_0(980)$ enlarges the symmetry energy of the infinite nuclear matter.
