Complement clauses, relative clauses, infinitivals, control, embedded and long-distance questions, long-distance and logophoric binding, and adverbial clauses are central topics of the grammar of well-studied languages. Furthermore, the analysis of complex sentences is also frequently revelatory for the analysis of less complex structures, as shown by, for example, German verb-second and verb-final structures. Descriptive work on understudied languages, however, rarely tackles complex sentences, and the tools to gather theoretically insightful data on complex sentences are underdeveloped. Project COSY addresses this gap in five indigenous languages of Brazil: Maxakalí, Kaingang (both Je), Yudja, Kawaiwete (both Tupi), and Kuikuro (Carib). The contribution of COSY consists of both theoretical insights concerning the syntactic and semantic principles and parameters of human language and practical advances for techniques to describe complex sentences in fieldwork description of languages. To accomplish its goals, COSY creates an intensive exchange between the Brazilian researchers at UFRJ and German researchers at ZAS.The theoretical goals of COSY concern the understanding of the areas mentioned at the beginning, and in addition also on switch reference marking, indexical shift, case systems, evidentiality, and speech-act marking. The close study of complex sentence structures in the five Amerindian languages is going to lead to new discoveries that are important for the syntactic and semantic theory within universal grammar. The practical advances these insights are based on are spontaneous elicitation techniques, an improved questionnaire, and several further experiments COSY develops to systematically gather data on complex sentences. As we refine these techniques, we simultaneously collect a unique corpus of data on five understudied and endangered Brazilian languages of three different language families. Finally, the project advances the adoption of leading technology in fieldwork data analysis and storage in cooperation with the Canadian, Dutch, German, and Brazilian researchers and institutions.