In this dissertation, Random Sets and Advanced Sampling techniques are combined for general and efficient uncertainty quantification. Random Sets extend the traditional probabilistic framework, as they also comprise imprecision to account for scarce data, lack of knowledge, vagueness, subjectivity, etc. The general attitude of Random Sets to include different kinds of uncertainty is paid to a very high computational price. In fact, Random Sets requires a min-max convolution for each sample picked by the Monte Carlo method. The speed of the min-max convolution can be sensibly increased when the system response relationship is known in analytical form. However, in a general multidisciplinary design context, the system response is very often treated as a “black box”; thus, the convolution requires the adoption of evolutionary or stochastic algorithms, which need to be deployed for each Monte Carlo sample. Therefore, the availability of very efficient sampling techniques is paramount to allow Random Sets to be applied to engineering problems. In this dissertation, Advanced Line Sampling methods have been generalised and extended to include Random Sets. Advanced Sampling techniques make the estimation of quantiles on relevant probabilities extremely efficient, by requiring significantly fewer numbers of samples compared to standard Monte Carlo methods. In particular, the Line Sampling method has been enhanced to link well to the Random Set representation. These developments comprise line search, line selection, direction adaptation, and data buffering. The enhanced efficiency of Line Sampling is demonstrated by means of numerical and large scale finite element examples. With the enhanced algorithm, the connection between Line Sampling and the generalised uncertainty model has been possible, both in a Double Loop and in a Random Set approach. The presented computational strategies have been implemented in the open source general purpose software for uncertainty quantification, OpenCossan. The general reach of the proposed strategy is demonstrated by means of applications to structural reliability of a finite element model, to preventive maintenance, and to the NASA Langley multidisciplinary uncertainty quantification challenge.