Uncertainty propagation has established itself as a fundamental area of research in all fields of science and engineering. Among its central topics stands the problem of modeling and propagating distributional uncertainty, i.e., the uncertainty about probability distributions. In this paper, we employ tools from Optimal Transport to capture distributional uncertainty via Optimal Transport ambiguity sets, which we show to be very natural and expressive, and to enjoy powerful topological, geometrical, computational, and statistical features and guarantees. We show that these ambiguity sets propagate nicely and intuitively through nonlinear, possibly corrupted by noise, transformations, and that in many cases the result of the propagation is again an Optimal Transport ambiguity set. Moreover, whenever this is not the case, we show that the result of the propagation can be tightly upper bounded by another Optimal Transport ambiguity set. Importantly, our methodology allows us to capture exactly how complex systems shape distributional uncertainty. To conclude, we exemplify our findings in three fundamental applications in forward propagation, inverse problems, and distributionally robust optimization.