In the domains of both inflection and derivation, there is evidence for both rules of exponence (which realize specific morphosyntactic properties or derivational categories through the introduction of specific morphological markings) and rules of composition (which determine how such rules of exponence apply in the definition of a compound’s inflected forms or derivatives). A single, general rule of composition accounts for the definition of a wide range of derivatives from compound bases; nevertheless, ordinal derivation demonstrates the considerable extent to which rules of composition may vary across languages. Evidence from a diverse range of languages is used to motivate a typology of ordinal derivation whose distinct types embody different rules of composition.