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What is the most descriptively and explanatorily adequate format for syntactic structures and how are they constrained? Different theories of syntax have provided various answers: sets, feature structures, tree diagrams... Building on formal and empirical insights from a wide variety of approaches spanning more than 70 years (including Transformational Grammar, Relational Grammar, Lexical-Functional Grammar, and Tree Adjoining Grammar), this monograph develops a new, mathematically grounded, framework in which objects known as graphs, and the constraints that follow from them, are argued to provide the best characterisation of the system of expressions and relations that make up natural language grammars. This new approach is motivated and exemplified via detailed and formally explicit analyses of major syntactic phenomena in English and Spanish.