This article is concerned with optimal control problems subject to a second order elliptic PDE and additional pointwise constraints on the gradient of the state. In particular, the existence of solutions on nonsmooth polygonal or polyhedral domains is analyzed. In this situation the solution operator for the PDE does not provide enough regularity to state the pointwise constraint for any right-hand side due to the appearance of singularities associated to the corners, edges, and vertices of the domain. Further, necessary optimality conditions for the solution of the optimization problem in two and three space dimensions are derived. However, in the three-dimensional case certain critical angles along the edges of the domain have to be circumvented in the derivation of the optimality conditions. Finally, the derived optimality conditions are utilized to deduce additional regularity for the control variable.