The magnetic field is the dominant force in the Sun's outer atmosphere, the corona. It determines the large-scale structure of the corona, governs the small-scale activity that heats the plasma, and powers energetic transient phenomena. The study of the coronal magnetic field is an important component of efforts to understand the corona as a whole, but the high plasma temperature and relative weakness of the field in the corona preclude direct measurement of the field in this region. Theoretical modeling, therefore, provides critical support to our investigation of the coronal magnetic field. This thesis contains an exploration of magnetic modeling in the context of the physics of the solar atmosphere. Using the novel stereoscopic capability of the Solar Terrestrial Relations Observatory, we compare the 3D coronal field structure with various models to study the distribution of the magnetic field and large-scale currents in several active regions. In addition to using existing models, we developed an original modeling approach: using several magnetic dipoles under the solar surface, configured based on comparison with the observed field structure, we overcome certain weaknesses inherent in existing models. The misalignment angles between the vectors of the reconstructed and model fields measure the departure of the model field from the observed field structure, indicating the non-potentiality of the coronal field. The dipole modeling method achieves a significant reduction in misalignment compared with previously established models, implying a closer agreement between our model field and observed field structures. We use the misalignment measurements to examine the free energy contained in active region magnetic fields and find that following major flares, the field relaxes substantially from a high-shear configuration to a low-shear configuration, indicating a significant loss of free energy in the field. The results in this thesis yield insight into the physics of the solar atmosphere and provide a means to better understand the complex region between the photosphere and corona. Improved understanding of this region will elucidate how the field connects the two regions, and the way in which energy is transported from the convective solar interior into the corona and heliosphere.