Numerical methods based on a parametric reformulation of such PDE problems emerged in the engineering literature in the 1990s as more efficient and rapidly convergent alternatives to Monte-Carlo sampling in cases where the dimension of the stochastic space is moderate (of the order of 10 random parameters). Recent research into these methods suggests that their advantageous approximation properties can best be achieved by using an adaptive refinement strategy, when spatial and stochastic components of the approximate solution are judiciously chosen in the course of numerical computation. The design of optimal adaptive algorithms remains an open question however. The proposed research programme aims at the design, theoretical analysis and efficient implementation of the state-of-the-art adaptive algorithms applicable to a range of PDE problems with random inputs.