The dynamics of Cooper pairs and vortices in a Josephson-junction array is investigated. For this purpose, a Hamiltonian is constructed in terms of vortex charges. Josephson-type equations for vortices are derived. A comparison with the Cooper-pair Hamiltonian shows that the roles of the magnetic field and induced charge density are reversed. The vortex and Cooper-pair Hamiltonians are approximately self-dual when EC/EJ = π2/2 (EC = e2/2C), which results in an array resistivity close to h/4e2.