We consider a class of parametric elliptic partial differential equations (PDEs), where the coefficients and the right-hand side depend on infinitely many (uncertain) parameters. For this class of PDEs, we present an adaptive multilevel stochastic Galerkin finite element method of the classical form (SOLVE -> ESTIMATE -> MARK -> REFINE). Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. The adaptive algorithm is proved to be convergent in the sense that the estimated error converges to zero. Under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces. This is joint work together with Alex Bespalov (U Birmingham) and Dirk Praetorius (TU Wien).