The authors discuss so called certification problems \(\mathbb{P}[G(X) \geq a] \leq \epsilon\), which means the problem of showing that with probabilty at least \(1-\epsilon\) a response function \(G\) of a given system will not exceed a given safety threshold \(a\). Since \(\mathbb{P}\) and \(G\) often not known a priori, they introduce the set \(\mathcal{A}\) of all admissible scenarios \((f,\mu)\) for the unknown reality \((G,\mathbb{P})\) and investigate the inequality \(\inf_{(f,\mu)\in\mathcal{A}} \mu[f(X) \geq a] \leq \mathbb{P}[G(X) \geq a] \leq \sup_{(f,\mu)\in\mathcal{A}} \mu[f(X) \geq a]\). Within this framework, the authors discuss the problem of uncertainty quantification methodologically and mathematically. Under certain assumptions, they prove reduction theorems which allow reduction to finite dimensional optimization problems. Further, using McDiarmid’s inequality and Hoeffding’s inequality, they present socalled optimal concentration inequalities. In some cases, they find nonpropagation of uncertainties. The results are applied in detail to practical examples e.g. to Small Particle Hypervelocity Impact Range facilities, to the Seismic Safety Assessment of Structures and to Transport in Porous Media.