Partial Differential Equations (PDEs) are fundamental in modeling phenomena across physics, engineering, finance, and biology. The Finite Element Method (FEM) is a powerful and flexible numerical technique for approximating solutions to PDEs, especially in complex geometries. However, as the dimensionality of the problem increases, FEM faces significant computational challenges due to the curse of dimensionality. This article provides a comprehensive overview of finite element methods for high-dimensional PDEs, discussing their mathematical foundations, discretization strategies, computational challenges, and recent advances—including hybrid approaches with machine learning. We highlight applications, current research directions, and the future outlook for FEM in tackling high-dimensional problems.