Two effects can often be observed when checking numerical simulation models against measured data from the described physical systems. On the one hand, a precise specification of all parameters in these models is often impossible for various reasons---sometimes, they do not even describe any physical quantities that could be measured. Moreover, many physical processes contain a natural degree of variability which cannot be described by deterministic models, not even by highly accurate ones. If actual measurements are to be used to calibrate the model, i.e. its unknown parameters, it is evident that noisy data will never allow an exact inference. This inexactness, in turn, implies that arbitrarily accurate statements, especially predictions, cannot be made about the behavior of the physical system. Such observations provide a direct motivation for uncertainty quantification, which aims to describe, quantify, and compute these very uncertainties. This thesis considers possibilistic methods for statistical inference with imprecise probabilities. In this sense, it belongs to a young subfield of statistics and uncertainty quantification which has its origins in the Dempster-Shafer theory of evidence and the theory of fuzzy sets by Zadeh. More precisely, it makes a contribution to the applied theory of imprecise probabilities by linking theoretical and numerical insights with practical applications in engineering and showing how possibilistic methods can be used profitably.