The interstellar medium – the space between the stars in our Galaxy – is multiphase, turbulent, and magnetic. Magnetism in the interstellar medium is difficult to observe and to simulate, and the study of interstellar magnetic fields is riddled with open questions. In this Thesis we make progress in several important areas. We use analytic theory, simulations, and observations to advance our understanding of an important plasma instability, of the diffuse neutral medium, and of prospects for uncovering cosmic inflation. We take an unusual approach to the study of the magnetorotational instability, the mechanism thought to be the primary driver of turbulence and angular momentum transport in astrophysical accretion disks. We conduct a weakly nonlinear analysis of the instability in several important geometries, and derive an envelope equation that governs the evolution of the system on long length- and timescales. We show that the saturated state of the magnetorotational instability may itself be unstable on these large spatial and temporal scales, and we demonstrate that the character of these instabilities will depend on the geometry of the background magnetic field.

We posit a possible new saturation mechanism for the magnetorotational instability in a local geometry, when a particular nonideal effect is considered. We derive new insights into the diffuse interstellar medium, where we present the discovery that thin, linear neutral hydrogen structures are ubiquitous in the cold neutral medium. We demonstrate that these linear features are extremely well aligned with the interstellar magnetic field, as traced by both starlight polarization and polarized dust emission. We discuss the implications of this discovery for cosmological studies. A major goal of modern cosmology is the detection of a particular signature in the polarized cosmic microwave background that would be direct evidence for inflation. This goal has thus far been thwarted by the polarized foreground emission from magnetically aligned interstellar dust grains. We demonstrate that the alignment of neutral hydrogen with the interstellar magnetic field can be used to produce higher-fidelity maps of the foreground polarization field, and we present and test a new Bayesian method for constructing improved foreground maps.