Neutral-atomic and molecular outflows are a common occurrence in galaxies, near and far. They operate over the full extent of their galaxy hosts, from the innermost regions of galactic nuclei to the outermost reaches of galaxy halos. They carry a substantial amount of material that would otherwise have been used to form new stars. These cool outflows may have a profound impact on the evolution of their host galaxies and environments. This article provides an overview of the basic physics of cool outflows, a comprehensive assessment of the observational techniques and diagnostic tools used to characterize them, a detailed description of the best-studied cases, and a more general discussion of the statistical properties of these outflows in the local and distant universe. The remaining outstanding issues that have not yet been resolved are summarized at the end of the review to inspire new research directions.