In this talk we discuss a so-called intrusive approach for the forward propagation of uncertainty in PDEs with uncertain coefficients. Specifically, we focus on stochastic Galerkin finite element methods (SGFEMs). Multilevel variants of such methods provide polynomial-based surrogates with spatial coefficients that reside in potentially different finite element spaces. For elliptic PDEs with diffusion coefficients represented as affine functions of countably infinitely many parameters, well established theoretical results state that such methods can achieve rates of convergence independent of the number of input parameters, thereby breaking the curse of dimensionality. Moreover, for nice enough test problems, it is even possible to prove convergence rates afforded to the chosen finite element method for the associated deterministic PDE . However, achieving these rates in practice using automated computational algorithms remains highly challenging, and non-intrusive multilevel sampling methods are often preferred for their ease of use. We discuss an adaptive framework that is driven by a classical hierarchical a posteriori error estimation strategy — modified for the more challenging parametric PDE setting — and present numerical results.