Next year, it will be 50 years since Marsden and Weinstein’s seminal paper on symplectic reduction. Perhaps surprisingly, this classic tool is all the more relevant today, when the interest of theoretical and mathematical physicists shifted towards higher codimensions. In this seminar I will review my recent work on the foundations of a theory of symplectic reduction for gauge theories on a manifold with boundaries and corners, which has numerous physical consequences for our understanding of soft charges and “electric flux superselection” at a null boundary. This work also provides a new perspective on the well-known and widely-used Ashtekar-Streubel phase space.