Abstract: As simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability, it has become critical to identify sources of model uncertainty as well as to quantify their effect on model outputs. For systems modeled by partial differential equations with random distributed parameters, statistical sampling methods such as Monte Carlo and stochastic collocation have proven both versatile and easy to implement. Multilevel sampling improves upon traditional sampling by dynamically incorporating the model's spatial discretization into the sampling procedure, thereby not only increasing efficiency but also allowing for a closer monitoring of overall convergence behavior. Originally developed for Monte Carlo sampling, these methods have since been extended to more general sampling methods, most notably stochastic collocation. We give a brief overview of the ideas underlying these methods and show how they can be used in the forward propagation of uncertainty and possibly to statistical inverse problems.