Computer simulations are a widely used tool in computational science and engineering to, e.g., analyze the behavior of components or materials, to enhance manufacturing processes with fast and accurate a-priori forecasts, or even to control those processes during ongoing operation. With the help of these tools, we wish to make reliable assertions and predictions for one or more quantity of interests (QoI), also in the presence of uncertainty, e.g., in process conditions, material properties or similar. Thus, methods from the field of Uncertainty Quantification (UQ) can enhance the quality of processes and products by augmenting the results for the QoI with quantified probability measures. We consider sampling-based UQ methods that usually require a great number of model evaluations. Thus, employing high-fidelity models may easily exceed available resources. Here, the usage of surrogate models, which are computationally cheaper, can provide a remedy. Therefore, we first follow the path of Model Order Reduction (MOR). In particular, we explore the benefits of intrusive MOR techniques and use Proper Orthogonal Decomposition (POD) with a subsequent Galerkin projection of the operators onto the constructed subspaces. As an alternative, we also investigate the advantages of Gaussian Process Regression (GPR) as a meta model. Finally, the integration of the resulting surrogate models into an UQ setting is demonstrated for applications coming from polymer processing.