This book presents a controlled evaluation of three widely practised syntactic theories on the basis of the extremely complex agreement system of Archi, an endangered Nakh-Daghestanian language. Even straightforward agreement examples are puzzling for syntacticians because agreement involves both redundancy and arbitrariness. Agreement is a significant source of syntactic complexity, exacerbated by the great diversity of its morphological expression. Imagine how the discipline of linguistics would be if expert practitioners of different theories met in a collaborative setting to tackle such challenging agreement data - to test the limits of their models and examine how the predictions of their theories differ given the same linguistic facts. Following an overview of the essentials of Archi grammar and an introduction to the remarkable agreement phenomena found in this language, three distinct accounts of the Archi data examine the tractability and predictive power of major syntactic theories: Head-driven Phrase Structure Grammar, Lexical Functional Grammar, and Minimalism. The final chapter compares the problems encountered and the solutions proposed in the different syntactic analyses and outlines the implications of the challenges that the Archi agreement system poses for linguistic theory.