This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order s ∈ (0, 1). There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of fractional PDEs with measure-valued datum. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first order optimality conditions. Notice that the adjoint equation is a fractional PDE with measure as the right-hand-side datum. We use the characterization of the fractional order dual spaces to study the regularity of the state and adjoint equations. We emphasize that the classical case (s = 1) was considered by E. Casas in [14] but almost none of the existing results are applicable to our fractional case.