We briefly review the goals of our project of investigating and clarifying the mathematical underpinnings of stochastic Galerkin methods, currently one of the key approaches for approximating the solution of PDEs with random data, as well as developing efficient computational solution methods. We will split our presentation between the two subprojects. The first, presented by Antje Mugler, will discuss a general approach to prove existence and uniqueness of the solution to variational formulations of PDEs with random data. A priori error bounds for the Galerkin solution and generalized polynomial chaos expansions are also addressed. The second, presented by Elisabeth Ullmann, will give a progress report on linear solvers for stochastic Galerkin discretizations of the lognormal diffusion problem which arises, for example in steady-state groundwater flow simulations. In this situation, we deal with random coefficients which are approximated via a nonlinear function in a fixed number of statistically independent Gaussian random variables. We discuss different formulations of the lognormal diffusion problem along with iterative solvers and preconditioning techniques for the associated Galerkin systems.