This is the first of two volumes arising from the INdAM Intensive Period "Poisson Geometry and Mathematical Physics," which took place in Naples from May 13 to July 12, 2024. These two volumes bring together contributions from numerous leading experts and young researchers in the field.They present the state-of-the-art of their research on topics such as: Lie groupoids and Lie algebroids; symplectic and Poisson geometry as well as other related geometries; Dirac geometry and moduli spaces of flat connections; reduction and geometric representation theory; and higher, derived, shifted and graded structures.These two volumes are primarily intended for researchers specializing in these fields.