The numerical solution of nonlinear elliptic partial differential equations (PDEs) is a cornerstone in computational science and engineering, underpinning models in physics, mechanics, and beyond. Adaptive finite element methods (AFEM) have emerged as a powerful tool for efficiently and accurately solving these challenging equations, especially when solutions exhibit singularities or sharp gradients. This article presents a comprehensive overview of the mathematical foundations, algorithmic strategies, and practical implementation of AFEM for nonlinear elliptic PDEs. We discuss the variational formulation, error estimation, mesh refinement strategies, and convergence properties. Numerical results and case studies illustrate the superior performance of adaptive methods compared to standard finite element approaches. The discussion highlights current challenges and future research directions in this rapidly evolving field.