Structure-preserving Galerkin methods have emerged as a powerful class of numerical discretizations that faithfully reproduce key mathematical and physical properties of partial differential equations at the discrete level. By preserving structures such as divergence constraints, exact sequences, energy dissipation, symmetries, and long-time dynamics, these methods often deliver enhanced robustness, stability, and physical fidelity. This minisymposium aims to bring together researchers working on the design, analysis, and implementation of structure-preserving Galerkin methods. Topics of interest include compatible finite element methods, discontinuous Galerkin methods, hybridizable discontinuous Galerkin methods, hybrid high-order methods, mixed finite element methods, pressure-robust and divergence-conforming discretizations and adaptive algorithms. Applications include incompressible flow problems subject to divergence constraints, Maxwell equations, elasticity, wave propagation, kinetic equations, and other multiphysics systems. Particular emphasis will be placed on recent advances in preserving fundamental analytical structures, including mass conservation, incompressibility, energy and entropy stability, and hypocoercivity, as well as on rigorous numerical analysis, efficient solvers, and adaptive methodologies.