Summary: We consider the inexact adaptive multilevel SQP-method from [J. C. Ziems and S. Ulbrich, SIAM J. Optim. 21, No. 1, 1–40 (2011; Zbl 1214.49027)] for the efficient solution of optimization problems governed by nonlinear Partial Differential Equations (PDEs) and propose an extension to control constraints. For this, we introduce a suitable criticality measure in the presence of control constraints as a natural generalization of the reduced gradient norm. We then obtain a fully adaptive inexact SQP-algorithm for optimal control problems with control constraints that starts on any coarse discretization of the infinite-dimensional problem and refines the meshes when and where necessary — without requiring a (suitable) a-priori given hierarchy of discretizations. The implementable refinement criteria are based on local error estimators and the criticality measure. A particular refinement condition for the control due to control constraints is included. The algorithm allows inexact linear system solvers and provides implementable accuracy requirements for iterative solvers of the linearized PDE and adjoint PDE on the current discretization. Appropriate decrease conditions are presented and their satisfiability under standard assumptions is shown. We prove global convergence to a stationary point of the infinite-dimensional problem. Numerical results that show the efficiency of this approach as well as the capability of defining the active set of the optimal control are presented.