In this review, the origins of astrochemistry and the initial applications of quantum chemistry to the discovery of new molecules in space are discussed. Furthermore, more recent successes and failures of quantum astrochemistry are explored. Finally, the application of quantum chemistry to the chemical study of space is driving developments in large-scale computational science. Consequently, cloud computing and large molecule computations are discussed. Astrochemistry is a natural application of quantum chemistry. The ability to analyze routinely and completely the structural, spectroscopic, and electronic properties of any given molecule, regardless of its laboratory stability, make this tool a necessary component for astrochemical analysis. The sizes of the computations scaling with the number of electrons and degrees-of-freedom can become limiting, but proper choices of methods can provide unique insights. The chemistry of the Earth is a small snapshot of the chemistries available in the universe at large, and the flexibility inherent within computation make quantum chemistry an excellent driver of new knowledge in fundamental molecular science as well as in astrophysics.