For the free massless spin-one and spin-two field theories one may write the action in a form which is manifestly invariant under electric-magnetic duality. This is achieved by introducing new potentials through solving the constraints of the Hamiltonian formulation. The price for making electric-magnetic duality invariance manifest through this direct procedure is losing manifest Lorentz invariance. Both theories admit supersymmetric extensions, which make the bosonic fields and their corresponding fermionic partners to be parts of the same geometrical object, a supermultiplet. We present in this paper the supersymmetric extension of the manifestly electric-magnetic duality invariant actions for the photon and the photino; and for the graviton and the gravitino. In each case the spinor fields transform under electric-magnetic duality in a chiral manner. For the spin-tree-half field, which possesses a gauge invariance, it is necessary to bring in a spinor "prepotential." As in previous cases the introduction of additional potentials to solve the constraints increases the number of gauge invariances of the action, thus keeping the number of degrees of freedom unaltered. The similarity in the formulations for the photon-photino and graviton-gravitino systems is remarkable.