During the early decades of the twentieth century the notion of a compact topological space arose as a generalization of results obtained in studies of the topology of the real line (in particular, the Heine-Borel theorem). Somewhat later, what is now called the Compactness Theorem for first-order logic was proved by Gödel as a lemma in his proof that every first-order axiom system is semantically complete. But for years thereafter connections between the two notions of compactness lay unrecognized and applications of compactness in logical contexts were not pursued. This talk will survey how the Compactness Theorem eventually came to be regarded as a fundamental tool in model theory and algebra, and will explore why recognition of it's usefulness was so long delayed.