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In this dissertation we have created theoretical models for finite nuclei, nuclear matter, and neutron stars within the framework of relativistic mean field (RMF) theory, and we have used these models to investigate the elusive isovector sector and related physics, in particular, the neutron-skin thickness of heavy nuclei, the nuclear symmetry energy, and the properties of neutron stars. To build RMF models that incorporate collective excitations in finite nuclei in addition to their ground-state properties, we have extended the non-relativistic sum rule approach to the relativistic domain. This allows an efficient estimate of giant monopole energies. Moreover, we have combined an exact shell-model-like approach with the mean-field calculation to describe pairing correlations in open-shell nuclei. All the ingredients were then put together to establish the calibration scheme. We have also extended the transformation between model parameters and pseudo data of nuclear matter within the RMF context. Performing calibration in this pseudo data space can not only facilitate the searching algorithm but also make the pseudo data genuine model predictions. This calibration scheme is also supplemented by a covariance analysis enabling us to extract the information content of a model, including theoretical uncertainties and correlation coefficients. A series of RMF models subject to the same isoscalar constraints but one differing isovector assumption were then created using this calibration scheme. By comparing their predictions of the nuclear matter equation of state to both experimental and theoretical constraints, we found that a small neutron skin of about 0.16 fm in Pb208 is favored, indicating that the symmetry energy should be soft. To obtain stronger evidence, we proceeded to examine the evolution of the isotopic chains in both oxygen and calcium. Again, it was found that the model with such small neutron skin and soft symmetry energy can best describe both isotopic chains, and the resultant values of the neutron-skin thickness and the symmetry energy are consistent with most current constraints. Finally, we addressed the recent tension between dense matter theory and the observation of neutron stars with rather small stellar radii. By employing Lindblom's algorithm, we were able to derive the underlying equation of state for assumed mass-radius relations having the "common radius" feature followed by recent analyses. We found that, in order to support two-solar-mass neutron stars, the typical stellar radii must be greater than 10.7 km—barely compatible with recent analyses—to prevent the underlying equation of state from violating causality.