  We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal
fully nonlinear elliptic equations with gradient constraints. We do not assume
any regularity about the constraints; so the constraints need not be $C^1$ or
strictly convex. We also obtain $C^{0,1}$ boundary regularity for these
problems. Our approach is to show that these nonlocal equations with gradient
constraints are related to some nonlocal double obstacle problems. Then we
prove the regularity of the double obstacle problems. In this process, we also
employ the monotonicity property for the second derivative of obstacles, which
we have obtained in a previous work.
