BREUDS is a research partnership between leading European and Brazilian research groups in dynamical systems, a prominent subject in mathematics. An extensive consortium of European and Brazilian institutions collaborates to provide world leading critical mass and support for research on the very forefront of the field. Work Packages reflect parallel priorities in the research. Transfer of knowledge is facilitated by conferences and five workshops. The project has excellent strategic value in view of the development of closer ties in higher education and research between the European Research Area and Brazil. The International Research Staff Exchange Scheme (IRSES) BREUDS has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 318999. In recent years and decades important progress has been made to understand the (generic) dynamics of low- dimensional (intrinsically autonomous, i.e. time-independent) dynamical systems, leading to the identification of new challenges, including high-dimensional dynamical systems (including PDEs and other types of functional differential equations), dynamical systems with particular structure (such as Hamiltonian systems, systems with symmetry, and systems on networks), and intrinsically nonautonomous dynamical systems (including dynamical systems with random elements). This joint research programme between Europe and Brazil relies on strong traditions in dynamical systems at both ends, and complementary expertise that will drive innovation and resolve open problems at the forefront of modern dynamical systems research. Typically for the multifaceted field of dynamical systems, the identified work packages have many places of contact among each other, and thus naturally form part of one scientific community, inspiring and challenging each other in many ways.