In this talk we will give a brief introduction to stochastic Galerkin methods, (a class of stochastic finite element methods) which facilitate the solution of elliptic PDEs where uncertainties are present in material parameters, boundary conditions and/or source terms. This is an alternative methodology to traditional Monte Carlo sampling methods. Probabilistic information about the system inputs is incorporated into the discretisation scheme in such a way that one linear system of equations is solved once. The methods will be illustrated with a simple groundwaterflow problem where the permeability coefficients of the porous medium are unknown. We discuss the efficient solution of the resulting linear systems and the computation of statistics of the solution variables.