Mathematical models involving partial differential equations (PDE) arise in numerous applications ranging from Natural Sciences and Engineering to Economics. Random and stochastic PDE models become very powerful (and sometimes unavoidable) extensions of deterministic models when a coefficient or forcing term cannot be adequately described by a single realization, e.g. due to their natural spatial or temporal variability, or the lack of information. This advantage comes at the price that stochastic models are usually significantly more difficult to solve and their numerical simulations result often in prohibitive computational costs. Feasible and reliable computer simulations require mathematically rigorous systematic studies of stability, convergence and efficiency of numerical algorithms and approximation methods.