The relentless demand for high-fidelity simulations in science and engineering has positioned numerical analysis as a cornerstone of modern applied mathematics, bridging abstract mathematical theory and computational practice. This review examines the foundational principles and recent advancements in numerical methods that underpin stability, convergence, and computational efficiency across diverse application domains. Beginning with an overview of discretization techniques—including finite difference, finite element, spectral, and emerging mesh-free methods—the discussion emphasizes the critical interplay between consistency, stability, and convergence as formalized by the Lax equivalence theorem and its extensions to nonlinear problems. Particular attention is devoted to rigorous stability analysis frameworks, including energy methods, summation-by-parts operators, and G-stability criteria for time-stepping schemes, alongside a priori and a posteriori error estimation strategy that enable adaptive refinement and uncertainty quantification. The exposition further explores the computational landscape of modern numerical solvers, highlighting innovations in sparse matrix technologies, preconditioned iterative methods, and parallel implementations that leverage multi-GPU architectures to achieve scalable performance. Applications in structural mechanics, fluid dynamics, and reduced-order modeling illustrate the practical realization of these theoretical constructs. The review concludes by identifying persistent challenges in multiscale coupling, structure-preserving algorithms, and data-integrated methodologies that define the frontier of numerical innovation, underscoring the indispensable role of mathematical rigor in advancing computational capability.