Machine learning is an essential tool for predicting complex phenomena in a variety of fields, such as engineering, medicine, and finance. In addition to concrete predictions provided by the model, knowledge about the uncertainty of these predictions is crucial for informed decision-making. Due to their inherent ability to provide probability distributions alongside their predictions, Gaussian processes are especially well-suited for uncertainty quantification. This thesis explores various approaches for uncertainty quantification in the context of Gaussian process models. First, we apply sensitivity analysis to examine how the different input variables contribute to uncertainty in the prediction. Second, we quantify the additional uncertainty arising from numerical approximations used in place of the full model. Finally, we calibrate the predicted probability distribution to align more accurately with the empirical probability distribution. As an illustrative case study, these uncertainty quantification approaches are applied to a Gaussian process model built to predict the traction force on the wheels of a vehicle driving on soft soil. The methodologies and insights this thesis provides may serve as a potential pathway for researchers and practitioners seeking to deepen their understanding of uncertainty in their Gaussian process models.