Phase-field models are widely used to capture interface dynamics in new materials, lithium batteries and biomolecular systems, etc. This minisymposium focuses on state-of-the-art high-order structure-preserving algorithms that rigorously preserve intrinsic physical structures of phase-field models. All contributions include complete theoretical analysis and practical numerical implementations. We intend to connect theorists and application-oriented computational scientists to address open challenges in constructing high-order, linear-decoupled, adaptive structure-preserving schemes for industrial multi-scale simulations.