The revolution in digital technology that has had so many obvious effects in recent decades has not spared the field of astronomy. It has led to an enormous improvement in astronomers' ability to study the "time domain," the expected and unexpected ways in which celestial objects change on timescales ranging from milliseconds to centuries. In the field of radio astronomy a variety of advances have led to a new breed of observatories that are orders of magnitude more efficient at surveying the sky than previous facilities. These new observatories produce data at prodigous rates, however, and require sophisticated analysis to take full advantage of their capabilities. With several major facilities coming online in the next few years, there is an urgent need to prove that terabytes of data can be reliably turned into genuine astrophysical results. This dissertation develops tools and techniques for coping with this challenge and applies them to data obtained with the Allen Telescope Array (ATA), a pioneering next-generation radio observatory located in Northern California. The ATA was built from the ground up to be a fast survey instrument, incorporating a suite of the new technologies that figure prominently in the new telescopes. I develop and describe miriad-python, a framework for the rapid development of interferometric analysis software that is used in a variety of ways in my subsequent research. I also present a robust software system for executing multiple observing campaigns cooperatively ("commensally") at the ATA. Data from the ATA are difficult to analyze due to nontraditional features such as a large instantaneous field of view; continuous coverage of a large, interference-prone frequency range; and broadband, movable feeds; I describe and implement several methods for coping with these challenges. This technical work is driven by the needs of a variety of astrophysical applications. I use broadband spectra of starforming galaxies to investigate the "calorimeter" interpretation of their cosmic ray energetics. The data are consistent with a recent hypothesis that the magnetic fields in these galaxies are stronger than traditionally thought. I use the survey capabilities of the ATA to conduct ASGARD, a large survey of the Galactic component of the dynamic radio sky, which has remained poorly-explored due to the limitations of previous obseratories and the technical challenges involved. I discuss in detail the methods used to analyze the data and provide catalogs, maps, completeness functions, and variability statistics. I map extended radio structures in Galactic fields and show how they can be subtracted from the data to simplify the search for transient Galactic sources. I limit the density of transient sources brighter than 10 mJy to be 100 MeV) ∼ 1.5 days later. This timing is inconsistent with standard inverse-Compton models, suggesting that multiple mechanisms may be responsible for the system's high-energy emission.