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The main advances regarding the deep connections between probability and possibility measurement uncertainty representation (but not the propagation) over the last decade are reviewed. They concern the following: the definition of a possibility distribution equivalent to a probability of one from its whole set of dispersion intervals about one point for all of the probability levels, the bridges with the conventional dispersion parameters, the representation of a partial probability knowledge owing to a maximum specificity principle better than the maximum entropy principle, and also probability inequalities. The use of a possibility representation for common measurement situations such as the description of measurement results, measurand estimation, and expression of a priori uncertainty information is illustrated and then discussed in view of their use in further processing (propagation and fuzzy inference systems). The conclusion highlights the interests of the possibility approach and points out some remaining issues.