We consider optimal control problems of elliptic PDEs on hypersurfaces Γ in Rn for n =2,3.The leading part of the PDE is given by the Laplace-Beltrami operator,which is discretized by finite elements on a polyhedral approximation of Γ.The discrete optimal control problem is formulated on the approximating surface and is solved numerically with a semi-smooth Newton algorithm.We derive optimal a priori error estimates for problems including control constraints and provide numerical examples confirming our analytical findings.