Parametric partial differential equations (PDEs) arise in several contexts, e.g., in optimization problems and in mathematical models with inherent uncertainties. In such PDEs, the differential operators usually depend on large, possibly infinite, sets of parameters, so that naive applications of standard numerical methods often lead to the so-called 'curse of dimensionality', a deterioration of the convergence rates and an exponential growth of the computational cost as the dimension of the space increases. In this talk, we give an overview of a specific numerical method for solving such PDEs, the stochastic Galerkin finite element method (SGFEM). First, we introduce a basic version of the method (we call it ’single-level’) for a standard high-dimensional parametric elliptic boundary value problem. Then, we discuss some of our recent results, focusing on a ‘multilevel’ approach, a posteriori error estimation, adaptivity and convergence with optimal algebraic rates. It turns out hat the proposed approaches help to mitigate the curse of dimensionality.