Many partial differential equations arising in geometry, continuum mechanics, and mathematical physics involve unknowns constrained to nonlinear manifolds. Examples include geometric evolution equations, nonlinear elasticity, shell and rod models, Landau-Lifshitz dynamics, harmonic and wave maps, and theories of liquid crystals and other constrained materials. Their numerical approximation poses significant challenges due to the interplay between geometric constraints, nonlinearity, and stability. This minisymposium focuses on the development and analysis of higher-order numerical methods for manifold-valued PDEs. Topics include high-order finite element and discontinuous Galerkin methods, geometric and structure-preserving time integrators, error analysis, adaptive algorithms, efficient nonlinear solvers, and applications to continuum mechanics, geometric evolution equations, and constrained physical models. The aim is to bring together researchers working on the mathematical foundations, algorithmic development, and practical implementation of advanced discretization techniques for manifold-valued problems, fostering interactions between numerical analysis, scientific computing, and applications.