High-order methods can provide high accuracy per degree of freedom and high arithmetic intensity that is well suited for modern computing architectures. Their adoption in industrial and large-scale simulations of partial differential equations (PDEs), however, remains limited by robustness issues for nonlinear, under-resolved, or shock-dominated problems and also by complexity of implementation. Recent developments in methods with provable properties, such as nonlinear stability, conservation, positivity preservation, and well-balancedness, provide a route to reliability. However, these properties often come at the expense of significant increase in computational cost. For these methods to become practical, such properties must be obtained without losing the efficiency that makes high-order methods attractive. The objective of this minisymposium is to bring together researchers addressing these challenges from complementary directions. Topics include entropy- and energy-stable schemes, positivity-preserving discretizations, limiting, shock capturing, adaptation, and performance-oriented implementations on modern computing platforms. We also seek contributions on high-order operator construction aimed at increasing accuracy per degree of freedom. These include construction of summation-by-parts (SBP) operators, e.g., Pad\'e-type, general-function space, and upwind SBP operators. Constructions that leverage structures such as collocation, sparsity, and tensor-product are also of interest. The overarching goal of the minisymposium is to clarify the trade-offs between accuracy, robustness, operator complexity, discretization cost, and hardware efficiency, and to identify practical directions for deploying robust and efficient high-order methods in industrial and large-scale PDE simulations.