NOTE: This is the version that was originally submitted. The (substantially modified and improved) published version can be found here: https://onlinelibrary.wiley.com/doi/full/10.1111/synt.12287 • • • DOI: 10.1111/synt.12287 • • • Two fundamental tasks of syntactic inquiry are to identify the elementary structure-building operations and to determine what properties they have and why. This article aims to bring us closer to those goals by investigating Merge. Two recent definitions of Merge are evaluated. It is argued that both have significant strengths but also some drawbacks, and that set-theoretic definitions of Merge in general face conceptual problems. It is proposed that Merge is not set-theoretic but graph-theoretic in nature: the syntactic objects it operates on and creates are (bare-phrase-structure-compliant) phrase-structure trees. Two new formal definitions of Merge are proposed and evaluated. One obeys the No-Tampering Condition but makes it unclear why Merge(α, β) satisfies only one selectional feature of α, not all of them. The other accounts for that observation but narrowly violates the No-Tampering Condition. The larger picture that emerges is one in which Merge is a graph-theoretic, not a set-theoretic, operation. • • • (N.B. “Set-theoretic” is used here — as in other work on these issues [e.g., Seely 2006:190, Krivochen 2023a,b,c] — as convenient shorthand for ‘narrowly set-theoretic, without involving graphs/trees’ [since graphs are themselves defined in terms of sets!].)