Many variational problems in continuum mechanics, including the Hellinger–Reissner formulation of linear elasticity, the velocity–stress formulations of incompressible and complex flows, and their magnetohydrodynamic and liquid-crystalline counterparts, are governed by a stress tensor whose algebraic structure is dictated by the balance of angular momentum. In the classical setting this manifests as the symmetry of the Cauchy stress, while in polar and micropolar media it manifests as a controlled non-symmetry. Recent work has shown that whether a discretization respects this structure exactly, rather than approximately, may drastically affect the accuracy of the computed stress: schemes enforcing the structure only weakly may return arbitrarily poor stress approximations under anisotropic constitutive laws, even in zero-stress configurations, whereas schemes enforcing it strongly deliver stress errors independent of the constitutive law. We have termed this property material robustness, in analogy with pressure robustness in incompressible flow. Strongly symmetric elements often come in two flavors: very low-order, or impractically high-order with supersmoothness. Constructing intermediate, but still high-order, spaces and appropriate solvers remains an active area of research that some of the talks will address. Moreover, circumventing the symmetry constraint via weak imposition also requires high-order methods to obtain accurate solutions for some constitutive laws. This minisymposium gathers recent advances on material robustness across theory, implementation, methodology, and applications: the unifying theory and its finite element tensor calculus interpretation; high-order frameworks such as TDNNS and Trefftz methods, and the solvers they require; and applications to thin domains, liquid crystals, magnetohydrodynamic equilibria, and complex fluids. Together, the talks position structure preservation as a unifying design principle for accurate, reliable, high-order discretizations throughout continuum mechanics. We envisage two to three sessions.