We study the giant resonances of stable and unstable nuclei in the framewor
k of the consistent relativistic random phase approximation built on the no
nlinear relativistic mean field (RMF) ground states. The isoscalar and isov
ector modes of giant resonances for stable nuclei, such as Pb-208, Sm-144,
Sn-116, Zr-90, Ca-40, O-16 and Ca isotope chains are investigated. The cont
ribution to the giant resonances from the Dirac sea states and the currents
of vector mesons are examined. The results show that the effects of the Di
rac sea states on the isoscalar giant modes are pronounced, but become weak
er for light nuclei, while the contributions of the isovector modes are neg
ligible. The few sets of parameterizations of nonlinear RMF model, which ar
e commonly used to give a good description of ground-state properties of fi
nite nuclei, could also well describe the nuclear dynamic properties-giant
resonances. For nuclei with the extreme value of N/Z, low-lying collective
excitations are found in isovector dipole modes, which are mainly due to th
e ph excitation of the weakly bound states near Fermi surface and the isosp
in mixture effect.