High-order numerical methods for ordinary differential equations have become indispensable tools in applied mathematics, computational science, and engineering applications where accuracy, stability, and computational efficiency are critical requirements. This article presents a comprehensive analysis of advanced numerical techniques for solving ordinary differential equations, with particular emphasis on stability theory, convergence properties, and error estimation mechanisms that govern the performance of high-order schemes. The study examines finite difference methods, Runge–Kutta approaches, multistep techniques, and spectral methods, investigating their mathematical foundations and computational characteristics. Special attention is devoted to error analysis frameworks that enable rigorous assessment of numerical accuracy and the trade-offs between computational cost and solution quality. Applications in engineering systems, physical modeling, dynamical systems analysis, and data-driven computational frameworks are explored to demonstrate the practical significance of these methods. The review addresses current challenges in numerical analysis, including stiffness handling, long-time integration accuracy, and adaptive strategy development. Future research directions emphasize the integration of high-order methods with machine learning approaches, development of structure-preserving algorithms, and enhancement of computational efficiency for large-scale systems. This work provides valuable insights for researchers and practitioners seeking to select and implement appropriate numerical methods for complex ordinary differential equations arising in real-world applications.