Many issues in uncertainty quantification, as they emerge from the perspective of large scale scientific computations of increasing complexity, involve dealing with stochastic versions of the basic equations modeling the phenomena of interest. A common reaction is to generate samples of solutions by choosing parameters randomly and computing solutions repeatedly. It is quickly realized that this is much too computationally demanding (but not entirely useless). Another common reaction is to do a sensitivity analysis by varying parameters in the neighborhood of regions of interest, leading to adjoint methods and computations that are not much more demanding than the basic one for which we want to find error bars. One does not have to be a sophisticated probabilist or statistician to realize that there is room for some interdisciplinary research here. My experience in studying waves and diffusion in random media motivated me to look into uncertainty quantification and to address some of the emerging issues. One such issue is the study of the propagation of shock profiles in random (turbulent) media. I will introduce this problem and analyze it from the point of view of large deviations, which is a regime that is particularly difficult to explore numerically. This problem is of independent interest in stochastic analysis and provides an example of how ideas from this theoretical research area can be used in applications. This is joint work with J. Garnier and T.W. Yang.