  Anatolii Mal’tsev first gave the compactness theorem in 1938 (for first-order logic of any signature), and used it in 1940/1 to prove several theorems about groups; this seems to have been the first application of model theory within classical mathematics. Leon Henkin and Abraham Robinson independently rediscovered the theorem a few years later and gave some further applications. The theorem fails badly for nearly all  infinitary languages.