Hyperbolic PDEs govern a wide range of models in geophysics and continuum mechanics, from shallow water flows to compressible gas dynamics, multiphase flows, and elasticity. Their numerical solution requires high-order schemes that remain robust in the presence of shocks, moving interfaces, and stiff source terms, while preserving, at the discrete level, the structural properties of the continuous models, such as entropy inequalities, positivity of some variables, etc. This minisymposium brings together recent developments on discontinuous Galerkin, finite volume, and finite element methods addressing these issues, including entropy-stable and invariant-domain-preserving discretizations, well-balanced schemes, all-Mach formulations, subcell finite volume techniques, shock capturing on fixed and moving meshes, and efficient implicit and semi-implicit time integration. The applications considered range from shallow-water, landslide, and thin-film models, to multiphase systems and hyperbolic reformulations of continuum mechanics. The contributions span the entire spectrum, from the construction and analysis of the schemes to their implementation and validation through demanding benchmarks.