  We consider obstacle problems for the Willmore functional in the class of
graphs of functions and surfaces of revolution with Dirichlet boundary
conditions. We prove the existence of minimisers of the obstacle problems under
the assumption that the Willmore energy with the unilateral constraint is below
a universal bound. We address the question whether such bounds are necessary in
order to ensure the solvability of the obstacle problems. Moreover, we give
several instructive examples of obstacles such that minimisers exist.
