Research problems on relations between Nijenhuis geometry and integrable systems

The paper surveys open problems and questions related to interplay between the theory of integrable systems with infinitely and finitely many degrees of freedom and Nijenhuis geometry. This text has grown out from preparatory materials for the series of research symposia and workshops on Nijenhuis geometry and integrable systems held at SMRI (Sydney) and MATRIX (Creswick) in February 2022 and at La Trobe University (Melbourne) and MATRIX in February 2024, and from the open problem sessions at these events. It includes both relatively simple questions to get familiar with the topic, as well as challenging problems that are of great importance for the field.