  We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type
and prove a number of results on the existence of solutions, directional
differentiability and optimal control of such QVIs. We give three existence
theorems based on an order approach, an iteration scheme and a sequential
regularisation through partial differential equations. We show that the
solution map taking the source term into the set of solutions of the QVI is
directionally differentiable for general data and locally Hadamard
differentiable obstacle mappings, thereby extending in particular the results
of our previous work which provided the first differentiability result for QVIs
in infinite dimensions. Optimal control problems with QVI constraints are also
considered and we derive various forms of stationarity conditions for control
problems, thus supplying among the first such results in this area.
