High-order accurate and nonlinearly stable numerical methods are especially efficient for hyperbolic or convection-dominated partial differential equation (PDE) models, which contain both singularities and complex smooth solution structures. This minisymposium focuses on recent developments and applications of high-order numerical methods, especially weighted essentially non-oscillatory (WENO) finite difference and finite volume methods, and discontinuous Galerkin (DG) finite element methods, for hyperbolic and convection-dominated problems. Topics include algorithmic efficiency for high-order accurate methods and high-dimensional problems; theoretical analysis of high-order schemes; positivity-preserving, well-balanced, and other structure-preserving approaches; novel time-discretization methods; effective methods for ensuring the nonlinear stability of high-order schemes; etc. This minisymposium is also dedicated to Professor Chi-Wang Shu on the occasion of his 70th birthday, recognizing the profound impact of his work and celebrating his achievements.