This talk presents recent advances on the development of globally convergent semismooth Newton methods in function space for solving a class of quasi-variationsl inequalities (QVIs) associated with elliptic partial differential operators and obstacle-type constraints. Due to the possible set-valuedness of the solution map, selection principles such as retrieving the minimal or maximal solution are discussed and analyzed. The latter also turn out to be useful in the context of optimal control of QVIs. Finally, data uncertainty is studied and associated stationarity characterizations for optimal control problems of the associated QVI are derived. The talk ends by a number of numerical illustrations.