The characterization of the void region in amorphous packings is crucial to understanding the behavior of many physical systems. We describe some of the recent advances in quantifying, exactly, the volume and surface area of cavities in packings of hard spheres. We also discuss the extension of these methods for the calculation of a precise geometrical free-volume distribution in the hard-sphere fluid. Accurate information regarding the statistical geometry of the dense hard-sphere fluid is obtained. The role of the void space in novel simulations involving metastable liquids is also discussed.