Since the introduction of the X-bar principles in syntax it is commonly assumed that adpositions are complement-taking heads of PPs. While this treatment is well motivated for many adpositions in many languages, there are some which - in some of their uses - would be better treated as nonhead daughters of verbal or nominal projections. These exceptional adpositions, which I will henceforth call 'minor', are the central topic of the article: I will provide criteria for their identification, propose a detailed treatment of the constructions which contain them, and address the question of whether they form a natural class. The wider significance of the analysis consists in the evidence which it provides against the wide-spread practice of treating function words as heads of phrasal projections. This practice, which is mainly associated with Chomsky (1986) and which has led to the postulation of a plethora of functional projections, including CP, DetP, and InflP, is not well suited to a treatment of the minor adpositions and should be replaced with an alternative in which these adpositions are treated as functor daughters in verbal or nominal projections. The analysis draws on various types of descriptive and formal grammar, as well as on corpus-based research, but the format in which it is cast is the constraint-based lexicalist one of Head-driven Phrase Structure Grammar; see Pollard and Sag (1994). A brief introduction to the HPSG notation is provided in section 2.2. The exemplification will be based on Dutch, but at various points comparisons will be made with English and German.