\indent The problem of the densest packing of space by congruent regular tetrahedra has a long history, starting with Aristotle's assertion that regular tetrahedra fill space, and continuing through its appearance in Hilbert's 18th Problem. This talk describes its history and many recent results obtained on this problem, including contributions from physicists, chemists and materials scientists. The current record for packing density is held by my former graduate student Elizabeth Chen, jointly with Michael Engel and Sharon Glotzer.