The fundamental goal of this paper is to argue for postulating that a clearly identifiable group of (underived) relational nouns has an argument structure in addition to what is often called a lexical conceptual structure - at least in languages like Hungarian. First, I posit relational nouns in a general typological setting, pointing out that it is necessary to attribute at least lexical conceptual structures (LCS) to them. Then I claim that Hungarian body part nouns like kéz `hand' have both inalienable and non-inalienable uses, which have become grammaticalized. In the former, they must be associated with an argument structure (AS), and in the latter, their LCS complement has been eliminated (hence, they have no argument structure, either). Next, I discuss the treatment of relational nouns of the szomszéd `neighbour' type, which behave partially differently. The essence of my analysis is that they also have ASs, and I account for the optionality of their possessor argument by invoking the standard suppression operation. I also show that there is independent motivation for assuming that Norwegian body part nouns also have ASs, although their behaviour is different from that of their Hungarian counterparts. Finally, I demonstrate that there are several (LFG) analyses of different phenomena in a variety of languages whose crucial assumption, shared by my approach in this paper, is that possessors are true syntactic arguments of relational nouns.