The design of complex vehicles, such as space launchers and hypersonic vehicles, often relies on physical models, called simulators, to predict their behavior. In this thesis, simulators take a vector input and outputs the system response, that is a function discretized on a mesh, with points located at the nodes or cells. Uncertainties may affect the simulator input, be it design-related (dimensions, mechanical properties of the materials, etc.) or a lack of knowledge of the environment (wind gusts, etc.). These uncertainties influence the output function, which then becomes random. The objective of uncertainty propagation is to measure that impact. Two metrics are considered in this thesis. The first is a functional α-quantile: for each location in the mesh, we take the smallest value such that the probability of being below it exceeds α. The second uses the notion of excursion set, defined as the set of mesh locations where the output exceeds a given threshold. Since the simulator output is random, so is the excursion set. The second metric is then a confidence region for that set, that is, a subset of the mesh whose probability of containing the excursion set exceeds a fixed threshold. Classical uncertainty propagation methods require generating a large number of realizations of the random functional output, obtained from running the simulator at a sample of the input variables. Since the simulator has a high computational cost, the approaches are impractical. To alleviate this issue, one can replace the simulator with a surrogate model, which is an approximation with a reduced computational cost. When the simulator output is functional, building the surrogate model consists of two steps: dimensionality reduction which projects the functional output onto a low-dimensional space, called the latent space, and interpolation or regression in that latent space. This thesis uses a combination of principal component analysis with Kriging, also called Gaussian process modeling. The surrogate model is trained from a set of realizations of the simulator inputs and outputs. Active learning strategies make the best use of the available training computational budget by selecting the training points based on the statistical measures to be estimated. This thesis develops two methods aiming to improve the estimation of functional quantiles and excursion-set confidence regions. Both leverage the predictive uncertainty model – the epistemic uncertainty, caused by a lack of knowledge. These methods are applied to multiple analytical and physical case studies, the latter in the context of aerospace vehicle design. In addition, different simulators with varying accuracy and computational cost are sometimes available. The surrogate model can exploit these multiple fidelity levels to reduce the evaluation cost and improve prediction accuracy. Recently, various multi-fidelity surrogate models with functional outputs have been proposed, with little to no comparison between them. This manuscript introduces a common framework encompassing all methods, enabling both theoretical comparisons and the construction of new variants, accompanied by a benchmark on case studies of increasing complexity.