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 definition of a possibility distribution equivalent to a probability one from its whole set of dispersion intervals for all the probability levels, the bridges with the conventional dispersion parameters, the representation of a partial probability knowledge thanks to a maximum specificity principle better than the maximum entropy principle and also related to probability inequalities. The use of a possibility representation for common measurement situations such as the description of measurement results, measurand estimation and the expression of a priori uncertainty information are illustrated, and then discussed in view of their use in further processing (propagation, fuzzy inference systems). The conclusion highlights the interests of the possibility approach and points out some remaining issues.