  This paper introduces the notion of state constraints for optimal control
problems governed by fractional elliptic PDEs of order $s \in (0,1)$. 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 \cite{ECasas_1986a} but
almost none of the existing results are applicable to our fractional case.
