Stochastic Galerkin (SG) methods provide a surrogate modelling technique for facilitating forward UQ in PDEs with uncertain inputs. Unlike conventional sampling methods, such as Monte Carlo, SG schemes yield approximations that are functions of the random input variables so that all realisations of the PDE solution are essentially approximated simultaneously. Since they use simple tensor product approximation spaces, standard SG schemes give rise to huge linear systems with coefficient matrices with a characteristic Kronecker product structure. Solving these systems is often a bottleneck when working on standard desktop computers. In this presentation, we will review two potential remedies. The first is to learn a lower-dimensional Galerkin approximation space using a bespoke a posteriori error estimator, leading to smaller linear systems that can be solved with modest computational resources. The second is to retain large tensor product spaces and recast the associated Kronecker system as a matrix equation. One can then apply low rank methods that iteratively construct a bespoke reduced basis and apply projection techniques to solve a problem of reduced size to determine a solution in factored form.