  In §1 the basic syntax and semantics of infinitary languages are laid down; their expressive power is then displayed by means of examples.  §2 is devoted to those infinitary languages which permit only finite quantifier sequences: these languages turn out to be relatively well-behaved. §3 is devoted to a discussion of the compactness problem for infinitary languages and its connection with purely set-theoretical questions concerning “large” cardinal numbers. In §4 an argument is sketched which shows that most “infinite quantifier” languages have a second-order nature and are, ipso facto, highly incomplete. §5 provides a brief account of a certain special class of sublanguages of infinitary languages for which a satisfactory generalization of the compactness theorem can be proved. This section includes a subsection on the definition of admissible sets. Historical and bibliographical remarks are provided in §6.