The heliosphere and solar magnetic field play an important role in protecting the solar system from harmful, high-energy Galactic radiation. Until recently, the magnetic field had been assumed to be passive, carried outwards by the solar wind. The influence of the solar magnetic field on the plasma has just begun to be understood. Among the consequences, the magnetic field could cause the heliotail to be short, collimating the flow into two lobes instead of the classical long, comet-like tail. In this dissertation, I investigate the role certain aspects of the magnetic field have on the heliosphere and detail how interstellar neutral particles alter its effect on the environment. From the observation by Voyager 1 (V1) and Voyager 2 (V2), it is clear that the plasma environment in the outer heliosphere is not fully understood. I present the first time-dependent model of the outer heliosphere that includes solar-cycle variations of the magnetic field strength. I find that the model can accurately predict the plasma environment at V2 but cannot describe all features observed at V1, suggesting additional processes are present. The effect of including the heliospheric current sheet (HCS) on large-scale modeling of the heliosphere is also studied. The inherent numerical dissipation in the HCS reduces the magnetic field strength in the heliosheath; however, the two-lobe structure of the heliotail remains. Neutral hydrogen has also been shown to greatly affect the location of the heliospheric boundaries. The large mean free path of these neutrals requires them to be described kinetically. To understand how the neutrals affect the influence of the solar magnetic field, I developed the Solar-wind with Hydrogen Ion Exchange and Large-scale Dynamics (SHIELD) model, a kinetic-magnetohydrodynamic model of the outer heliosphere. The model couples a 3D Monte-Carlo model to the magnetohydrodynamic solver. SHIELD reproduces the results of similar models, namely a higher filtration of neutrals into the heliosphere when compared to a fluid description of the atoms. When SHIELD is applied to the heliotail, the two-lobe structure persists even with kinetic neutrals. These results show that the solar magnetic field plays a crucial role in determining the heliospheric structure.