Subsrface models are used to forecast the behavior of the earth regarding fundamental or applied questions as encountered for example in petroleum reservoir management. These models and the associated interpretations often obey the principle of parsimony, which tells that the simplest model able to explain observations should be considered. This quest for simplicity is certainly useful to identify trends and make sense out of the various measurements made about the subsurface. However, this practice also raises some questions. Indeed, incomplete spatial coverage, limited resolution and data processing errors raise ambiguities during interpretation. Would it be useful to capture the associated uncertainties? Do they have an impact on forecasts? Did the interpretation miss some important unseen features? In the flow simulation community, many facts suggest that these questions should be answered positively. This raises another set of questions about how to come up with several possible models representing what is seen, poorly seen, possible and at what scale. In this talk, I will review some recent work addressing these questions in the field of three-dimensional (3D) structural modeling. First, I will present some results about the impact of structural uncertainties in the case of a poorly imaged segmented normal fault for three physical problems: stress field estimation, flow simulation, wave propagation and gravity response. Then, I will discuss some uncertainties related to the spatial layout of faults and stratigraphic formations. Depending on the data at hand, these uncertainties may relate to the position of faults and horizons, their lateral extent and the connectivity of fault networks. In some instances, they also relate to the internal structure of faults and to their existence. To sample these uncertainties, prior knowledge must be formulated in probabilistic terms and a new breed of object-based simulation is needed to reflect the interactions between geological structures. I will illustrate this by showing some examples of recently developed 3D stochastic structural simulators, and will discuss avenues for further improvements of these simulators and challenges ahead.