We model the uncertainties in (random) coefficient functions of an elliptic partial differential equation by expanding these coefficients as function series with scalar random coefficients. This gives us a deterministic formulation of a random PDE. Due to the combination of stochastic and spatial unknowns, this gives us a high-dimensional elliptic PDE. We present an adaptive stochastic Galerkin method for solving this PDE and discuss the optimality of this method. The method combines a multilevel representation of stationary random fields with a residual-based spatial adaptive scheme. An optimal operator compression is used for the stochastic operator. A Bramble-Pasciak-Xu (BPX)-frame is used to obtain a residual estimate and to achieve appropriate error reduction in the iterative linear solver. The numerical results and in the wavelet-case a complete rigorous analysis show that the obtained scheme is optimal. This talk is based on joint work with M. Bachmayr, M. Eigel and H. Eisenmann.