In the literature on prepositions, a simple binary distinction between ``lexical\'\' and ``functional\'\' uses is commonly assumed. In this thesis I investigate the nature of the lexical vs. functional distinction for prepositions, and I argue that these two uses correspond to cardinal points of a spectrum of prepositional uses. This spectrum can be modelled descriptively as the interaction of two properties: form-fixedness and perceived meaningfulness. At the functional corner of the spectrum, prepositions are characterized by low meaningfulness and high fixedness, while at the lexical corner, prepositions have high meaningfulness and low fixedness. There are also, however, prepositions that are perceived to be both meaningful and fixed, and these present a problem for the notion of a simple binary lexical vs. functional dichotomy. <br />A number of empirical tests have been proposed for inducing a binary classification of prepositional uses---for example, formation of the pseudopassive and wh-questions, and specifier attachment. While these are all interesting phenomena individually, they do not converge on a single classification collectively, and I conclude that there are no broad generalizations to be captured by postulating a primitive lexical vs. functional distinction theoretically.<br />My own analysis, formalized in the framework of Head-Driven Phrase Structure Grammar, includes two binary distinctions between contentful vs. empty prepositions, and between prepositions with syntactically selected form vs. those without. The interaction of these distinctions results in an idealized representation of the three-cornered descriptive spectrum of meaningfulness and fixedness. I discuss various ways in which my more or less discrete formal representations can give rise to gradient behavior on a descriptive level.